The theory of scaling called finite similitude does not involve dimensional analysis and is founded on a transport-equation approach that is applicable to all of classical physics. It features a countable infinite number of similitude rules and has recently been extended to other types of governing equations (e.g., differential, variational) by the introduction of a scaling space \(\Omega _{\beta }\), within which all physical quantities are deemed dependent on a single dimensional parameter \(\beta \). The theory is presently limited to physical applications but the focus of this paper is its extension to other quantitative-based theories such as finance. This is achieved by connecting it to an extended form of dimensional analysis, where changes in any quantity can be associated with curves projected onto a dimensional Lie group. It is shown in the paper how differential similitude identities arising out of the finite similitude theory are universal in the sense they can be formed and applied to any quantitative-based theory. In order to illustrate its applicability outside physics the Black-Scholes equation for option valuation in finance is considered since this equation is recognised to be similar in form to an equation from thermal physics. It is demonstrated that the theory of finite similitude can be applied to the Black-Scholes equation and more widely can be used to assess observed size effects in portfolio performance.
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1 Introduction
The theoretical cornerstone of modern scaled experimentation is a theory that was established over a century ago and involves representing the governing equations in dimensionless form involving the arrangement of physical properties into dimensionless groups. Buckingham [1] was the first to recognise that dimensioned relationships possess hidden dependencies that could be removed by making the relationships dimensionless. The methodology called dimensional analysis is well-known, being relatively easy to use, and has the advantage that it is able to characterise dominant physics through a subset of the dominant Pi groups [1‐3]. A negative feature, as regards scaling, is its dependence on an invariance principle that seldom features in practical systems of even moderate complexity [4, 5]. The issue is one of scale effects, and by definition dimensional analysis offers no solution in their presence, which consequently significantly limits its reach in afflicted scaled experiments [3]. Although, the lack of similarity in experimentation is undoubtedly limiting this does not mean that scaled experiments have no value. An alternative approach to scaling is through the application of sophisticated computational models [6], which to a large extent have superseded the need for scaled experimentation. The downside of modelling however is inherent uncertainties in boundary and material data, and possibly uncertain physics, all of which can limit the predictive capability of the computational approach.
Scaled models and experiments provide an investigative approach that is accessible, cost-effective and time-saving. The biggest drawback is undoubtedly scale effects, which can have the effect of undermining the usefulness of any results obtained at scale [7]. Changes in behaviour that manifest with scale can be so significant that the approach can be totally undermined. It is readily apparent from the ad hoc changes made to scaled models reported in the literature that scale effects are an unwelcome affliction. A prime example is the application of shake-table experimentation to investigate the behaviour of buildings in earthquake investigations, where ad hoc changes typically feature. The adding of slabs of mass to counter scale effects is one such modification [8] but such a solution has drawbacks however since slabs can shift to produce irregular motion [9‐12]. An alternative approach to adding mass is the modification of gravity [13] through the application of a centrifuge, revealing the extraordinary effort afforded to counter scale effects. Other systems recognised to suffer from scale effects are electromechanical systems [14] where changes with scale for micro-electromechanical systems (MEMS) is the focus on the book by Baglio et al. [15]. The response with scaling of actuation mechanisms featuring in MEMS (e.g., electrostatic, magnetic, thermal, piezoelectric) is examined by Liu and Bar-Cohen [16]. An obvious effect are scale dependencies arising from changes in geometric measures since length scales linearly, area quadratically and volume cubically. Although scale dependencies do not necessarily infer scale effects it is evident that they play a significant role although to fully understand the source of scale effects it is required to understand how scaling influences the governing constraining laws on a system. The scaling of Maxwell equations for a single scaled experiment has been examined by Gustafson [17] and the works of Sinclair [18] and Pries and Hofmann [19], with both thermal physics and electromagnetism featuring in the latter study. Scaled experiments can play a role in a set-up known as Hardware-In-the-Loop (HIL) simulation, which is an approach that is designed to facilitate the testing of embedded systems. The current focus on electrification and hybrid systems [20‐23] to meet climate challenges has elevated the importance of HIL simulation for testing purposes. It is appreciated that scaled experimentation can feature in HIL simulation forming the hardware component and consequently providing the means to perform realistic simulation in a cost-efficient manner [24]. As pointed out in reference [24] however the fundamental concern with scaled hardware is the presence of scale effects. Although the software component can to some degree mitigate against the presence of scale effects there exists limits to this form of correction. Presently the only possible corrective action available is the involvement of dimensional analysis and ad hoc scaling rules [25]. Despite the obvious advantage of HIL simulation the added uncertainly brought by scale effects to the investigation is evidently disadvantageous.
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For the past century the similitude rule provided by dimensional analysis [26] (i.e., dimensionless equations being invariant under scaling) was believed to be unique with no other similitude rules existing. The reason for this is all too apparent since the breaking of this invariance principle means scale effects with the full and scaled systems behaving differently. However, a new way of looking at scaling has recently appeared in the open literature, that presents scaling in an entirely new light and addresses the question of uniqueness of similitude rules. The new approach called finite similitude is unusual in the sense that it is underpinned by a metaphysical concept termed space scaling that cannot be practically achieved. Although space scaling cannot be realised practically it can be defined mathematically, which provides the means to assess the effect of metaphysical scaling on the governing constraining equations. These equations are presented in integral-transport form to describe behaviour on a control volume (i.e., a region of space) since this form is immediately impacted by space scaling. This new approach to scaling has been applied to a range of practical problems ranging from metal forming, powder processing to structural and biomechanics [27‐32]. A feature of the finite similitude approach is the projection of the governing transport equations in a trial space (where the scaled experiment resides) onto the physical space (where the full-scale experiment resides). This projection is critical to the whole approach since it has the effect of quantifying all scale dependencies, some revealed explicitly (e.g., volume and area) and others implicitly (e.g., scalar, vector and tensor fields) [33‐35]. The projected space is called \(\Omega _{\beta }\) and can be regarded as a metaphysical representation of the real physical attributes of spaces in which experimentation takes place, as such it is possible to define all of classical mathematical physics on this space. A feature of \(\Omega _{\beta }\), as a metaphysical space, is the ability to relate physics in experimental spaces (trial or physical) using the aforementioned finite similitude rules. Furthermore, its dimensional equivalence to the physical space \(\Omega _{ps}\) serves as the foundation of a theoretical extension of the finite similitude scaling approach to all of science involving governing equations of any type (e.g., pdes, odes, integral, transport, least action, variational, and stoichiometric equations). The focus and novelty of this work is the connection between scale invariance methods, dimensional analysis and high-order finite similitude theory viewed initially through the physical laws constraining Newtonian mechanics, but then beyond, to finance with special interest in higher order similitude rules to highlight the reach of the approach.
The foundation of the finite similitude theory including metaphysical space scaling is recapped in brief in the first part of Sect. 2. The focus is on isotropic scaling which maintains geometric similarity although anisotropic scaling is indeed possible but discussed elsewhere [36]. The theory begins with the identification of inertial frames for the physical space (within which the full-scale experiment sits) and in the trial space (within which the scaled experiment sits) and defining of a space-scaling map between the frames to quantify the expansion and contraction of space. The interrelationship between fixed and moving control volumes as defined in previous work [27] is also briefly recapped in this section. The projected space \(\Omega _{\beta }\) is presented in Sect. 2.1 along with relationships for moving control volumes in the trial and physical spaces. Shown in this section is how a transport equation on \(\Omega _{\beta }\) formed from the projection of an arbitrary transport equation in the trial space is exactly of the form expected if the space \(\Omega _{\beta }\) were a real physical space. This observation is key to treatment of \(\Omega _{\beta }\) as a physical space and the direct application of alternative mathematical physical formulations along with the connection to dimensional analysis. Although the projection from the trial space \(\Omega _{ts}\) effectively defines the physics on the space \(\Omega _{\beta }\), similitude rules are required to establish links to the physical space \(\Omega _{ps}\). The differential-similitude rules and their integration is discussed in Sect. 2.2, where identities for one, two and three scaled experiments are presented. These are respectively termed zeroth, first and second order finite-similitude rules with zeroth order being equivalent to dimensional analysis in the sense that proportional fields are involved in both cases. Newtonian mechanics is assessed in Sect. 3 in both transport and differential form along with the connection to dimensional analysis in subsection 3.2. It is shown in Sect. 4 how dimensional analysis can be extended to replicate finite similitude theory and consequently extending the reach of finite similitude to all quantitative-based sciences. As a practical demonstration of this extension the scaling theory is applied to the Black-Scholes equation in Sect. 5. Additionally, in this section, the finite similitude theory is applied to work presented in the literature, examining size-effect influence on portfolio performance. The paper ends with a list of conclusions.
2 A brief recap on finite similitude
Prior to introducing the new formulation is beneficial to recap the basic tenets of the finite similitude theory, which was introduced by Davey et al. [27] and has been applied subsequently to many practical problems. These problems include those related to impact mechanics [30, 31, 33], dynamics [34, 37], earthquake mechanics [35], biomechanics [29, 38], powder compaction [28], metal forming [39], electromagnetism [24], fracture and fluid mechanics [41, 42]. The starting point for the finite similitude theory is space scaling, an imagined continuous single-parameter process, involving a positive real parameter \(\beta \). Space scaling is quantified mathematically by relating coordinate coefficients in the physical space \(x^i_{ps}\) (where the full-size process resides) to those in the trial space \(x^i_{ts}\) (where the scaled experiment sits) through a linear map \(\textbf{x}_{ps} \mapsto \textbf{x}_{ts}\). Explicitly, in differential terms, space scaling is the relationship \(d\textbf{x}_{ts} = \beta d\textbf{x}_{ps}\), which in component form is \(d{x}^i_{ts} = \beta d{x}^i_{ps}\). In Newtonian physics, the absolute times in the two spaces are related in a similar fashion, by a differential identity map of the form \(dt_{ts} =g dt_{ps}\), where g is positive, temporally and spatially invariant function of \(\beta \). Under the assumption that the inertial frames in the two spaces are orthonormal, then the scaling is isotropic with contraction identified with \(\beta < 1\), and no scaling returned on setting \(\beta =1\), and expansion provided by \(\beta >1\). The approach can readily be adapted for anisotropic space scaling, but this aspect is not considered further here being limited to niche applications [36].
Fig. 1
Relationships between moving control volumes \(\Omega ^*_{ts}\) and \(\Omega ^*_{ps}\) in their respective spaces tracking a moving gear train
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With the focus on space, and in view of their immediacy to changes in geometric measures (e.g., volume, area) control volume-based formulations are a central feature of the finite-similitude theory. All laws of nature in classical physics can be described in transport form, on a moving control volume, which identifies a moving region of space that is of peculiar interest. With scaling, the motion of control volumes in the trial and physical spaces can be expected to be related in some way, and specific details can be found in references [33‐35]. The basic idea is to describe the motions of the control volumes in the trial and physical spaces by means of velocity fields \(\textbf{v}_{ts}^*\) and \(\textbf{v}_{ps}^*\), respectively. The motion of the moving control volumes \(\Omega ^*_{ts}\) and \(\Omega ^*_{ps}\), are connected to these velocity fields by temporal derivatives \(\tfrac{D^*}{D^* t_{ts}}=\tfrac{\partial }{\partial t_{ts}} \big |_{\mathbf {\chi }_{ts}}\) and \(\tfrac{D^*}{D^* t_{ps}}=\tfrac{\partial }{\partial t_{ps}} \big |_{\mathbf {\chi }_{ps}}\), where \(\mathbf {\chi }_{ts}\) and \(\mathbf {\chi }_{ps}\) are coordinate points in reference control volumes \(\Omega ^{*ref}_{ts}\) and \(\Omega ^{*ref}_{ps}\), respectively. The overall concept is depicted in Fig. 1, where the motions of \(\Omega ^*_{ts}\) and \(\Omega ^*_{ps}\) are gauged relative to stationary reference control volumes \(\Omega ^{*ref}_{ts}\) and \(\Omega ^{*ref}_{ps}\), respectively. Their motion is readily described mathematically as the solutions to the differential equations \(\tfrac{D^* \textbf{x}^*_{ts}}{D^* t_{ts}}=\textbf{v}_{ts}^*\) and \(\tfrac{D^* \textbf{x}^*_{ps}}{D^* t_{ps}}=\textbf{v}_{ps}^*\), where \(\textbf{x}^*_{ts}\) and \(\textbf{x}^*_{ps}\) are points attached to their respective control volumes. Frobenius’s theorem guarantees unique solutions to these differential equations for regular velocity fields \(\textbf{v}_{ts}^*\) and \(\textbf{v}_{ts}^*\) and meaningful initial conditions.
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It is evident that the motion of the two control volumes depicted in Fig. 1 must be related and, in view of the scaling map \(d\textbf{x}_{ts} = \beta d\textbf{x}_{ps}\), it appears reasonable to assume that the identity \(d\textbf{x}^*_{ts} = \beta d\textbf{x}^*_{ps}\) applies relating points on the moving control volumes; clearly \(d\mathbf {\chi }_{ts} = \beta d\mathbf {\chi }_{ps}\). Additionally, in view of the temporal relationship \(d{t}_{ts} = g d{t}_{ps}\), it follows that the control volume velocity fields are related by the relatively simple relationship \(\textbf{v}_{ts}^*=g^{-1} \beta \textbf{v}_{ps}^*\). Note that the identity \(d\textbf{x}^*_{ts} = \beta d\textbf{x}^*_{ps}\) provides the measure relationships \(d{V}^*_{ts} = \beta ^3 d{V}^*_{ps}\) and \(d\mathbf {\Gamma }^*_{ts} = \beta ^2 d\mathbf {\Gamma }^*_{ps}\) for volume and area respectively, where \(d\mathbf {\Gamma }^*_{ts} = \textbf{n}^*_{ts}d{\Gamma }^*_{ts}\) and \(d\mathbf {\Gamma }^*_{ps} = \textbf{n}^*_{ps}d{\Gamma }^*_{ps}\), where \(\textbf{n}^*_{ts}\) and \(\textbf{n}^*_{ts}\) are unit vectors. Although the theory presented in this section has been widely applied it is nevertheless useful to draw a sharper distinction between what is actually happening in the physical space and what the similitude identities infer is happening. This can be achieved with the definition of the projected space \(\Omega _\beta \).
2.1 The projected physical space \(\Omega _{\beta }\)
Accepting that the space scaling relationship \(d\textbf{x}_{ts} = \beta d\textbf{x}_{ps}\) applies, the starting point in the formation of \(\Omega _\beta \) are the identities \(d{t}_{\beta } = g^{-1}d{t}_{ts}\) and \(\textbf{v}_{\beta }^*=g \beta ^{-1} \textbf{v}_{ts}^*\), where the subscript \(\beta \) infers the possibility of \(\beta -\)dependence. These two identities generalise the identities above (i.e., \(d{t}_{ps} = g^{-1} d{t}_{ts}\) and \(\textbf{v}_{ps}^*=g \beta ^{-1} \textbf{v}_{ts}^*\)) and give rise to the space \(\Omega _{\beta }\), which is similar in form to \(\Omega _{ps}\). The difference between \(\Omega _{\beta }\) and \(\Omega _{ps}\) is that the former permits a wider range of control volume motions, and the latter is restricted to satisfy \(\textbf{v}_{\beta }^*=\textbf{v}_{1}^*=\textbf{v}_{ps}^*\) (with \(\textbf{v}_{1}^*=\textbf{v}_{\beta =1}^*\)). This feature is not however the focus of the work presented here as it will be revealed that the space \(\Omega _{\beta }\) is physically realisable to all classical physics. It is convenient and always possible to set \(dt_{\beta }=dt_{ps}=dt_1\), i.e., arrange for \(t_{\beta }\) to be independent of \(\beta \) and define \(t_{\beta }\) to be the time-scale in the physical space. This follows because \(d t_{ts}\) being dependent on \(\beta \) can be set equal to \(g(\beta )dt_{1}\) and consequently \(d{t}_{\beta } = g^{-1} d{t}_{ts}=g^{-1} g d{t}_{1}=d{t}_{1}\).
Fig. 2
Relationships between moving control volumes \(\Omega ^*_{ts}\), \(\Omega ^*_{ps}\) and \(\Omega ^*_{\beta }\) in their respective spaces
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The velocity field \(\textbf{v}_{\beta }^*\) dictates the motion of the control volume \(\Omega _{\beta }^*\) in \(\Omega _{\beta }\) and consequently the identity \(d\textbf{x}_{\beta }^*=\textbf{v}_{\beta }^* dt_{\beta }\) applies, but substitution of \(d{t}_{\beta } = g^{-1} d{t}_{ts}\) and \(\textbf{v}_{\beta }^*=g \beta ^{-1} \textbf{v}_{ts}^*\) provides \(d\textbf{x}_{\beta }^*= \beta ^{-1}\textbf{v}_{ts}^*dt_{ts}=\beta ^{-1}d\textbf{x}_{ts}^*\). In other words, \(d\textbf{x}_{ts}^*=\beta d\textbf{x}_{\beta }^*\) and the following geometric relationships apply: \(d{V}^*_{ts} = \beta ^3 d{V}^*_{\beta }\) and \(d\mathbf {\Gamma }^*_{ts} = \beta ^2 d\mathbf {\Gamma }^*_{\beta }\), where \(d\mathbf {\Gamma }^*_{\beta } = \textbf{n}^*_{\beta }d{\Gamma }^*_{\beta }\), and where \(\textbf{n}^*_{\beta }\) is a unit vector. The connectivity between the reference and moving control volumes in the spaces \(\Omega _{ps}\), \(\Omega _{ts}\) and \(\Omega _{\beta }\) are provided in Fig. 2. Note the dashed arrow between control volumes \(\Omega _{ps}^*\) and \(\Omega _{\beta }^*\), which indicates that the relationship is undefined because of the unknown relationship between \(\textbf{v}_{ps}^*\) and \(\textbf{v}_{\beta }^*\). The simplest relationship between these two velocity fields is the one outlined in the previous section, which is returned on setting \(\textbf{v}_{\beta }^*=\textbf{v}_{1}^*=\textbf{v}_{ps}^*\). It is useful at this point to introduce the general transport equation for the trial space, which is of the form
where \(\mathbf {\Psi }_{ts}\) is the primary field of interest, \(\textbf{v}_{ts}\) is material velocity, \(\rho _{ts}\) is material density, and \(\textbf{J}_{ts}^\psi \) and \(\textbf{b}_{ts}^\psi \) are flux and source terms, respectively.
Substitution of \({\text {d}}{V}^*_{ts} = \beta ^3 {\text {d}}{V}^*_{\beta }\), \(d\mathbf {\Gamma }^*_{ts} = \beta ^2 d\mathbf {\Gamma }^*_{\beta }\), \({\text {d}}{t}_{ts} = g {\text {d}}{t}_{\beta }\), and multiplication throughout by g and a scalar \(\alpha _0^\psi \) (an assumed function of \(\beta \)) provides
Equation (2) is a projection of Eq. (1) on \(\Omega _{\beta }\) with no approximation involved, so the physics constrained to satisfy Eq. (1) on \(\Omega _{ts}\) are played out on \(\Omega _{\beta }\). Equation (2) is assumed to apply at any value of \(\beta \) and in particular applies at \(\beta =1\), which is the physical space which necessitates \(\alpha _{0}^\psi (1)=1\) and \(g(1)=1\), along with \( \rho _{1} \mathbf {\Psi }_{1}=\rho _{ps} \mathbf {\Psi }_{ps}\), \(\textbf{v}_{1}^*=\textbf{v}_{ps}^*\), \(\textbf{v}_{1}=\textbf{v}_{ps}\), \(\textbf{J}_{1}^\psi = \textbf{J}_{ps}^\psi \), and \( \rho _{1} \textbf{b}_{1}^\psi =\rho _{ps}\textbf{b}_{ps}^\psi \). A critical yet readily apparent feature of Eq. (2) is that it is identical in form to an arbitrary transport equation in any physical space. Consequently, in fairly broad terms, the space \(\Omega _{\beta }\) is able to be endowed with physics and physical formulations of different forms. Thus, it is possible to represent physics with the full spectrum of mathematical formulations including ordinary and partial differential equations, least action forms. Unlike traditional spaces however \(\Omega _{\beta }\) provides a connection between the trial space \(\Omega _{ts}\) and the physical space \(\Omega _{ps}\). The connection between the spaces \(\Omega _{\beta }\) and \(\Omega _{ps}\) is formularised through similitude rules that are assumed to apply to all physical quantities that reside in \(\Omega _{\beta }\) for which details are provided below.
2.2 Similitude rules and integration for transport equations
All classical physics is constrained by the transport equations introduced in the previous section being of the form \(\alpha ^\psi _0 T^\psi _0=0\), where \(\psi \) is a label used to differentiate between the different equations under consideration. In the finite similitude theory, differential operations are performed on the projected transport equations Eq. (2), which in effect dictate how these equations change with \(\beta \). The simplest rule, which can be shown to be equivalent to dimensional analysis, assumes that no change with \(\beta \) takes place. This can be expressed succinctly by the mathematical identity
where the sign "\(\equiv \)" denotes equality of the integrands to zero, for the pertaining balance law, making the derivative of the transport equations identically zero, which means that the left-hand-side of Eq. (3) vanishes under the derivative.
This rule that was initially introduced in reference [36] and its application explored in references [27‐32], and is termed zeroth-order finite similitude. Equation (3) is seldom satisfied in practical systems, but alternative rules involving higher derivatives are possible and the following definition is central to the finite-similitude approach [43]:
\(\forall \beta >0\) is a similitude rule (termed \(k^{th}\)-order finite similitude), where \(\alpha ^\psi _k:\mathbb {R}^+\mapsto \mathbb {R}\), \(\alpha ^{\psi }_k(1)=1\) and \(\alpha ^\psi _0 T^\psi _0=0\) is Eq. (2).
Definition 2.1 was initially introduced in reference [34] and is designed to capture the basic requirements of similitude rules, which include nesting, where lower-order similitude conditions if satisfied infer the satisfaction of higher order rules. The approach would cease to be of practical value without this feature and observe that zeroth-order finite similitude under this definition is \(T^{\psi }_{1} \equiv 0\) and first-order finite similitude is \(T^{\psi }_{2} \equiv 0\) and second order finite similitude is \(T^{\psi }_{3} \equiv 0\). Observe further that the first-order identity has two derivatives
and similarly for higher order, with the number of derivatives matching the number of scaled experiments involved.
The similitude identities defined by Eq. (4) can be readily integrated using a divided-difference table but require the application of a mean-value theorem to return exact similitude rules. The particular similitude rule defined by Eq. (3), integrates readily over the limits \(\beta _{1}\) and \(\beta _0\), but in term of divided difference takes the form
where a mean-value theorem is applied to provide an exact identity with \(\beta _1 \le \beta ^0_1 \le \beta _0\), and note that \(\alpha ^\psi _0 T^\psi _0(\beta _0)\equiv \alpha ^\psi _0 T^\psi _0(\beta _1)\) confirms that the projected transport equation does not depend on \(\beta \).
First-order integration is concerned with the solution of Eq. (5), which is achieved by considering an additional scale \(\beta _{2}\) and a similar divided-difference identity to Eq. (7), i.e.,
with \(\beta _2 \le \beta ^1_2 \le \beta _1\), where as above, a mean-value theorem is again utilised to provide an exact identity.
The first-order assumption means that \(\alpha ^\psi _1T^\psi _1(\beta ^0_1) \equiv \alpha ^\psi _1T^\psi _1(\beta ^1_2)\), which on substitution of Eqs. (7) and (8) provides after a little rearrangement
and since \(\alpha ^\psi _1\) is an indeterminate function of \(\beta \), it is understood that \(R^\psi _1\) takes the form of a parameter.
Second-order integration is concerned with the solution of Eq. (6), which follows on from first order by considering an additional scale \(\beta _{3}\) and a similar divided-difference identity to Eqs. (7) and (8), i.e.,
with \(\beta _3 \le \beta ^2_3 \le \beta _2\), where as above, a mean-value theorem is again utilised to provide an exact identity. Given the identities Eqs. (7), (8) and (11) and recalling that \(\alpha ^\psi _2T^\psi _2=\alpha ^\psi _2\tfrac{d}{d\beta }(\alpha ^\psi _1T^\psi _1)\) the following exact divided expression can be formed:
where \(\beta _3^2 \le \beta ^1_3 \le \beta _2^1\) and \(\beta _2^1 \le \beta ^0_2 \le \beta _1^0\). Given that second-order finite similitude requires \(\alpha ^\psi _2T^\psi _2(\beta ^1_3) \equiv \alpha ^\psi _2T^\psi _2(\beta ^0_2)\) it follows from substitution of Eqs. (7), (8) and (11) into this identity and after a significant degree of algebraic manipulation that,
and as above \(\alpha ^\psi _2\) is an indeterminate function of \(\beta \), making \(R^\psi _2\) a parameter.
It is evident that higher-order relationships can be returned by following an identical approach involving additional scales and divided differences. The rules have recently been shown to be unique in the sense that any alternative rule of the same order can be represented [43].
3 Mechanics in \(\Omega _\beta \)
The transport equations pertinent to mechanics can be immediately written in standard form on the space \(\Omega _{\beta }\). The equations of interest are the conserved equations for volume, mass, and momentum, and the non-conserved equation for movement (first introduced in reference [44] to constrain the displacement field). The finite-similitude approach focuses on the fields of interest and these equations constrain the fields of control-volume velocity \(\textbf{v}_{\beta }^*\), material density \(\rho _{\beta }\), material velocity \(\textbf{v}_{\beta }\), stress \(\mathbf {\sigma }_{\beta }\), body force density \(\rho _{\beta } \textbf{b}_\beta \), and displacement \(\textbf{u}_\beta \). The transport equations on \(\Omega _{\beta }\) take the form:
The form taken by Eq. (15) is obtained on setting the scalars to \(\alpha _0^1=\beta ^{-3}\), \(\alpha ^v_0=g\beta ^{-1}\alpha ^\rho _0\) and \(\alpha ^u_0=\beta ^{-1}\alpha ^\rho _0\), where along with g are assumed to be functions of \(\beta \); it is also assumed also that \(t_\beta =t_1=t_{ps}\) making \(t_\beta \) independent of \(\beta \). Since \(\textbf{v}_{\beta }^*\) is the velocity that dictates the movement of the control volume \(\Omega _{\beta }^*\) in \(\Omega _{\beta }\) it is apparent that a significant simplification occurs if \(\textbf{v}_{\beta }^*\) is independent of \(\beta \), i.e., \(\tfrac{{\text {d}}\textbf{v}_{\beta }^*}{{\text {d}}{\beta }}\equiv \textbf{0}\). This is a zeroth-order condition, and with \(\textbf{v}_{\beta }^*=\textbf{v}_{1}^*=\textbf{v}_{ps}^*\), it follows that \(\Omega _{\beta }^*=\Omega _{ps}^*\) and \(\tfrac{{\text {d}}}{{\text {d}}{\beta }}(\alpha _{0}^1 T_0^1)\equiv 0\) applies, i.e., Eq. (15a) satisfies the zeroth-order similitude rule Eq. (3). The conditions \(\Omega _{\beta }^*=\Omega _{ps}^*\) and \(t_\beta =t_{ps}\) allow the derivative similitude rules in Sect. 2.2 to apply directly to the integrands in the transport equations (15). Thus, two approaches are possible for the transport equations, i.e., the direct application of the differential identities or alternatively the application of the integrated forms in Sect. 2.2. The latter approach is particularly convenient for linking scaled experiments and is the approach adopted in all previous studies (see for example references [27‐32]). The types of relationships returned by this approach are presented in Table 1. Note that additional field relationships not explicitly appearing in the transport equations can be deduced, where in this case in Table 1 acceleration \(\textbf{a}_{\beta }=g^2 \beta ^{-1} \textbf{a}_{ts}\) is included. This is a feature of the finite-similitude approach where all fields are determinable and no recourse to constitutive equations is required. Observe that only one parameter \(R_1\) (also \(R_{1,1}\), \(R_{1,2}\) and \(R_2\) value) is present in Table 1 arising because the velocity field is present in all transport equations apart from the volume equation, where no first or second-order terms appear. Note that \(R_1\) (also \(R_{1,1}\), \(R_{1,2}\) and \(R_2\)) is a parameter as a consequence of the function \(\alpha _1=\alpha _1^\rho =\alpha _1^v=\alpha _1^u\) (also \(\alpha _2=\alpha _2^\rho =\alpha _2^v=\alpha _2^u\)) being an unknown function of \(\beta \).
3.1 Differential mechanics
It is of interest to examine how field-differential identities can be applied directly, where each field (\(\mathbf {\Psi }_\beta \), \(\mathbf {\Theta }_\beta \) say) satisfies one of the following identities \(\tfrac{d}{{\text {d}}\beta }\mathbf {\Psi }_\beta \equiv \textbf{0}\), \(\tfrac{d}{{\text {d}}\beta }(\alpha _1\tfrac{d}{{\text {d}}\beta }\mathbf {\Psi }_\beta )\equiv \textbf{0}\) or \(\tfrac{d}{d\beta }(\alpha _2\tfrac{d}{{\text {d}}\beta }(\alpha _1\tfrac{d}{{\text {d}}\beta }\mathbf {\Psi }_\beta ))\equiv \textbf{0}\), which respectively denotes zeroth, first and second order, although second-order relationships are restricted here to products of first-order terms. It is relatively easy to prove that if \(\mathbf {\Psi }_\beta \) is zeroth order and \(\mathbf {\Theta }_\beta \) is first or second order, then \(\mathbf {\Theta }_\beta \mathbf {\Psi }_\beta \) is also first or second order, respectively. Additionally, the product \(\mathbf {\Theta }_\beta \mathbf {\Psi }_\beta \) is second order if \(\mathbf {\Theta }_\beta \) and \(\mathbf {\Psi }_\beta \) are first order with \(\alpha _2=\alpha _1\). To see this consider the second order identity
which vanishes on differentiation with respect to \(\beta \).
This result provides the means to scale mechanics equations in any form, which includes the classical Newtonian, Lagrangian and Hamiltonian approaches and can also shed light on the scaling of empirical laws. All these formulations can be applied in the space \(\Omega _\beta \) and subsequently tested with the order identities to ascertain the requirements for scaling with one, two or three scaled experiments. Classical Newtonian mechanics for mass-spring-damper systems for example are governed in \(\Omega _\beta \) by ordinary differential equations of the form
where \([M_\beta ]\), \([C_\beta ]\) and \([K_\beta ]\) are mass, damping and stiffness matrices, respectively and \((u_\beta )\) and \((F_\beta )\) are column vectors for displacement and force.
The scaling of Eq. (17) is constrained by its form necessitating that each term individually (i.e., \((F_\beta )\), \([M_\beta ](\ddot{u}_\beta )\), \([C_\beta ](\dot{u}_\beta )\) and \([K_\beta ](u_\beta )\)) satisfies one of the order identities, but observe additionally that \((u_\beta )\), \((\dot{u}_\beta )\) and \((\ddot{u}_\beta )\) are of identical order (since \(dt_\beta =dt_1\)). If, for example, \((u_\beta )\) is first order and the whole system is required to be second order, then the matrices \([M_\beta ]\), \([C_\beta ]\) and \([K_\beta ]\) cannot be higher than first order. In this situation \((F_\beta )\) can be any order up to second order, although for a second-order system at least two of the terms (i.e., \((F_\beta )\), \([M_\beta ](\ddot{u}_\beta )\), \([C_\beta ](\dot{u}_\beta )\)) in Eq. (17) must be second order.
To illustrate the new \(\Omega _\beta \) approach and highlight issues that arise in scaling consider the simple damped vibratory system depicted in Fig. 3.
Fig. 3
Single degree damped-vibratory system in \(\Omega _{\beta }\) and free-body diagram showing forces arising from gravity, friction, spring, and the damper
×
An attractive feature of space \(\Omega _\beta \) is that systems can be approached in a traditional manner and in this case the governing equation is
where \(m_\beta \) is mass, \(c_\beta \) is the damping coefficient, \(k_\beta \) is stiffness, \(\mu _\beta \) is the friction coefficient, \(G_\beta \) is acceleration due to gravity, \(\theta _\beta \) is slope angle, and additionally, displacement \(u_\beta =\beta ^{-1}u_{ts}\), velocity \(v_\beta =\dot{u}_\beta =g\beta ^{-1}\dot{u}_{ts}\), and acceleration \(a_\beta =\dot{v}_\beta =\ddot{u}_\beta =g^2\beta ^{-1}\ddot{u}_{ts}\).
Equation (18) is projected from the trial space and it is necessary to understand the physical constraints imposed by the real system. Replica scaling is assumed here with materials unchanged in any scaled model apart possibly for the damper whose hydraulic fluid may require substitution. Additionally, it is assumed that the friction coefficient and slope angle remain unchanged, i.e., \(\mu _{\beta }=\mu _{ts}=\mu _{ps}\) and \(\theta _\beta =\theta _{ts}=\theta _{ps}\), respectively. The following relationships from the projection are readily deduced \(m_\beta =\alpha _{0}^\rho m_{ts}\), \(k_\beta =\alpha _{0}^\rho g^2 k_{ts}\), \(c_\beta =\alpha _{0}^\rho g c_{ts}\) and \(G_\beta =g^2\beta ^{-1}G_{ts}\). The density relationship \(\rho _{\beta }=\alpha _{0}^\rho \beta ^3 \rho _{ts} = \rho _{ps}\) requires \(\alpha _{0}^\rho =\beta ^{-3}\) and consequently \(m_\beta =\beta ^{-3} m_{ts}\), \(k_\beta =\beta ^{-3} g^2 k_{ts}\), and \(c_\beta =\beta ^{-3} g c_{ts}\). These are projected relationships, but a problem arises on replica scaling of the spring and damper, which behave as \(k_{ts}=\beta k_{ps}\) (with \(g=\beta \)) and \(c_{ts}=\beta c_{ps}\) (with \(g=\beta ^2\)), respectively (see reference [37] for details). Both \(g=\beta \) and \(g=\beta ^2\) cannot of course be set and the best choice is \(g=\beta \), which provides \(k_\beta =\beta ^{-1}k_{ts}= k_{ps}\) and \(c_\beta =\beta ^{-2}c_{ts} \not = c_{ps}\). However, by means of a fluid change in the damper (see reference [37]) the damper can be arranged to satisfy \(c_\beta =\beta ^{-2}c_{ts} = c_{ps}\), which is zeroth order as are \(m_\beta =m_{ps}\) and \(k_\beta =k_{ps}\). Despite this convenience however Eq. (18) is not zeroth order since on the right side \(G_\beta =g^2\beta ^{-1}G_{ts}=\beta G_{ts} \not = G_{ps} = 9.81 \mathrm {m/s^2}\). A single scaled model (despite a fluid change) cannot replicate the behaviour of the full scale model in Fig. 3. A two-scaled approach is obtained on making \(G_\beta \) (also \(u_\beta \) by default) first order, which requires \(\tfrac{d}{d\beta }(\alpha _1\tfrac{d}{d\beta }G_\beta )\equiv 0\) or equivalently \(\tfrac{d}{d\beta }(\alpha _1\tfrac{d}{d\beta }\beta )G_{ts}\equiv 0\) (since \(G_{ts}\) is constant), which is satisfied with \(\alpha _1=1\). Additionally, substitution of \(G_\beta =g^2\beta ^{-1}G_{ts}\) in the first-order relationship for acceleration in Table 1 gives,
and with the requirement that \(G_{ps}=G_{1}=G_{ts1}=G_{ts2}\), provides \(R_1=(1-\beta _1)(\beta _1-\beta _2)^{-1}\).
The situation where the hydraulic fluid cannot be changed is more challenging however and here a second-order solution must be considered. Recall that \(c_\beta =\beta ^{-2}c_{ts}\) but under the assumed physical constraint \(c_{ps}=\beta ^{-1}c_{ts}\), it follows that \(c_\beta =\beta ^{-1}c_{ps}\), so first order for \(c_\beta \) requires \(\tfrac{d}{d\beta }(\alpha _1\tfrac{d}{d\beta }c_\beta ) = \tfrac{d}{d\beta }(\alpha _1\tfrac{d}{d\beta }\beta ^{-1})c_{ps}\equiv 0\) which is satisfied with \(\alpha _1=\beta ^2\). The left hand side of Eq. (18) is second order with \(c_\beta \) and \(u_\beta \) first order but a problem remains because Eq. (16) requires \(\alpha _2=\alpha _1\) but \(\tfrac{d}{d\beta }(\alpha _2\tfrac{d}{d\beta }(\alpha _1\tfrac{d}{d\beta }G_\beta ))\not \equiv 0\) with \(\alpha _2=\alpha _1=\beta ^2\), effectively ruling out a three-experiment solution. Although the system in Fig. 3 is relatively simple it does nonetheless reveal the types of practical considerations involved in applying the finite similitude theory. The benefit of the space \(\Omega _\beta \) is demonstrated in the example presented here where similitude relationships are readily obtained between the experimental spaces (trial or physical) and \(\Omega _\beta \), for any \(\beta \), since physical laws and constraints apply in the same manner as in experimental spaces. The space \(\Omega _\beta \) fulfils an important role when the connection to scaled-dimensional invariance is examined, thus attention turns to the link between similitude theory and dimensional analysis.
3.2 The connection to dimensional analysis
The finite similitude theory makes no recourse to dimensional analysis but it is of interest nevertheless to examine this connection. In mechanics the theory has revealed that the fields \(\textbf{v}_{\beta }^*=g \beta ^{-1} \textbf{v}_{ts}^*\), \(\rho _{\beta }=\alpha ^\rho _0\beta ^3\rho _{ts}\), \(\textbf{v}_{\beta }=\beta ^{-1}g\textbf{v}_{ts}\), \(\textbf{u}_{\beta }=\beta ^{-1}\textbf{u}_{ts}\), \(\mathbf {\sigma }_{\beta }=\alpha ^v_0g\beta ^2\mathbf {\sigma }_{ts}\), and \( \rho _{\beta }\textbf{b}^v_{\beta }=\alpha ^v_0g\beta ^3\rho _{ts} \textbf{b}^{v}_{ts}\) satisfy the identity in Sect. 2.2. It is possible to show however that dimensional considerations facilitate the establishment of these relationships. Although a dimension of a physical quantity can be considered to be a function that assigns a scalar relating different units quantifying the physical quantity [3], a somewhat broader definition is applied here. Any property measurable in the broad sense that it can be quantified is deemed to posses a dimension. Dimensions form a group (i.e., the dimensional group) and are related through a multiplicative binary operation. An immediate advantage of this formalisation (over the units one) is that "dimensionless" quantities such as probability, number of shares, size of data sets, amongst others, can possess a dimension. A common system of units is assumed for the different physical space-time manifolds considered below but it is necessary to make a distinction between "common" dimensions in the different manifolds, which is a facet not available to the units approach [3]. These considerations are for purpose of extending the applicability of scaled-dimensional approaches and the formation of dimensionless groups [1] is not a consideration. Additionally, the dimension of scalars, vectors and tensorial quantities are defined by the dimension of their components regardless of the number of components involved.
Pure dimensions in physics for space, time, mass, temperature and charge are allocated capital letters L, T, M, \(\Theta \) and Q, respectively. The square bracket notation is used here to identify the dimension of a quantity, i.e., \(L=[\textbf{x}]\), \(T=[t]\) for space and time respectively. In particular, on making a distinction between dimensions in different space-time manifolds, \(L_{ts}=[\textbf{x}_{ts}]\) & \(T_{ts}=[t_{ts}]\), \(L_{ps}=[\textbf{x}_{ps}]\) & \(T_{ps}=[t_{ps}]\), and \(L_{\beta }=[\textbf{x}_{\beta }]\) & \(T_{\beta }=[t_{\beta }]\). In terms of dimensional analysis the finite similitude theory provides therefore the relationships \([\textbf{v}_{\beta }^*]=g \beta ^{-1} [\textbf{v}_{ts}^*]\), \([\rho _{\beta }]=\alpha ^\rho _0\beta ^3[\rho _{ts}]\), \([\textbf{v}_{\beta }]=\beta ^{-1}g[\textbf{v}_{ts}]\), \([\textbf{u}_{\beta }]=\beta ^{-1}[\textbf{u}_{ts}]\), \([\mathbf {\sigma }_{\beta }]=\alpha ^v_0g\beta ^2[\mathbf {\sigma }_{ts}]\), and \( [\rho _{\beta }\textbf{b}^v_{\beta }]=\alpha ^v_0g\beta ^3[\rho _{ts} \textbf{b}^{v}_{ts}]\). An important question here is whether the relationships for mechanics can be deduced from dimensional considerations alone, along with the identity \(\alpha ^v_0=g\beta ^{-1}\alpha ^\rho _0\). To answer this note first the kinematic terms with \([\textbf{v}_{ts}^*]=L_{ts}T_{ts}^{-1}\), \([\textbf{v}_{ts}]=L_{ts}T_{ts}^{-1}\) and \([\textbf{u}_{ts}]=L_{ts}\) but since \(L_{ts}=\beta L_{ps}\) and \(T_{ts}=g T_{ps}\) it follows that \(g \beta ^{-1} [\textbf{v}_{ts}^*]=L_{ps}T_{ps}^{-1}\), \(g \beta ^{-1} [\textbf{v}_{ts}] =L_{ps}T_{ps}^{-1}\) and \(\beta ^{-1}[\textbf{u}_{ts}]=L_{ps}\). It is evident that the objective here is to return the dimensions of the trial space to the physical space, which is equivalent to an attempt to remove the effect of space scaling. Consider further the kinetic terms with the appreciation that another scalar is involved with the property mass, i.e. \(\alpha _{0}^\rho M_{ts}=M_{ps}\) it follows that \(\alpha ^\rho _0\beta ^3[\rho _{ts}]=M_{ps}L_{ps}^{-3}\), as required. Similarly, the stress term \(\alpha ^v_0g\beta ^2[\mathbf {\sigma }_{ts}]\) on substitution of \(\alpha ^v_0=g\beta ^{-1}\alpha ^\rho _0\) gives \(\alpha ^\rho _0g^2\beta [\mathbf {\sigma }_{ts}]\) and since \([\mathbf {\sigma }_{ts}]=M_{ts}L_{ts}^{-1}T_{ts}^{-2}\) consequently \(\alpha ^\rho _0g^2\beta [\mathbf {\sigma }_{ts}]=M_{ps}L_{ps}^{-1}T_{ps}^{-2}\). Similarly, \(\alpha ^v_0g\beta ^3[\rho _{ts} \textbf{b}^{v}_{ts}]\) on substitution of \(\alpha ^v_0=g\beta ^{-1}\alpha ^\rho _0\) returns \(\alpha ^\rho _0g^2\beta ^2[\rho _{ts} \textbf{b}^{v}_{ts}]\) but \([\rho _{ts} \textbf{b}^{v}_{ts}]=M_{ts}L_{ts}^{-2}T_{ts}^{-2}\), so \(\alpha ^\rho _0g^2\beta ^2[\rho _{ts} \textbf{b}^{v}_{ts}]=M_{ps}L_{ps}^{-2}T_{ps}^{-2}\). The analysis confirms that \([\textbf{v}_{\beta }^*]=[\textbf{v}_{ps}^*]\), \([\textbf{v}_{\beta }]=[\textbf{v}_{ps}]\), \([\rho _{\beta }]=[\rho _{ps}]\), \([\textbf{u}_{\beta }]=[\textbf{u}_{ps}]\), \([\mathbf {\sigma }_{\beta }]=[\mathbf {\sigma }_{ps}]\), and \( [\rho _{\beta }\textbf{b}^v_{\beta }]=[\rho _{ps} \textbf{b}^{v}_{ps}]\). Thus confirming that the dimensions on \(\Omega _{\beta }\) and \(\Omega _{ps}\) are identical, but of course, in no way ensuring that numerical values match. This is achieved on satisfying the zeroth-order identity for each field (e.g., \(\tfrac{d}{{\text {d}} \beta }\textbf{v}_{\beta }\equiv \textbf{0}\)).
4 Extended dimensional analysis
In view of the link between finite similitude and dimensional analysis it of interest to examine the connection more formally. The dimensions form a group called the dimensional group [45] but here a distinction is made between groups and dimensions in each space, which are denoted \(\mathcal {G}_{ps}\), \(\mathcal {G}_{ts}\) and \(\mathcal {G}_{\beta }\). Members of the group \(\mathcal {G}_{ts}\) for mechanics are the identity \(1_{ts}\), time \(T_{ts}=[t_{ts}]\), space \(L_{ts}=[\textbf{x}_{ts}]\), mass \(M_{ts}=[m_{ts}]\) along with algebraic multiplicative combinations such as \([\textbf{x}_{ts}]^{1/2}[t_{ts}]\), \([{t}_{ts}]^{-2}[m_{ts}]^{2}\), and so on. The members of \(\mathcal {G}_{ps}\) look identical to \(\mathcal {G}_{ts}\) apart from a change in subscript, i.e., "ps" replaces "ts". The members of \(\mathcal {G}_{\beta }\) are formed from quantities defined on the space \(\Omega _{ts}\), where the relationships \(\beta ^{-1}L_{ts}=L_{ps}\), \(g^{-1}T_{ts}=T_{ps}\) and \(\alpha _{0}^\rho M_{ts}=M_{ps}\) apply. In essence, \(\mathcal {G}_{ps}\) and \(\mathcal {G}_{\beta }\) coincide on setting \(L_{\beta }=\beta ^{-1}L_{ts}\), \(T_{\beta }=g^{-1}(\beta )T_{ts}\) and \(M_{\beta }=\alpha _{0}^{-\rho }(\beta ) M_{ts}\) with the members of \(\mathcal {G}_{\beta }\) being algebraic multiplicative combinations of the identity \(1_{\beta }\), length \(L_{\beta }\), time \(T_{\beta }\) and mass \(M_{\beta }\). Note that although the spaces \(\Omega _{\beta }\) and \(\Omega _{ps}\) are identical from a dimensional viewpoint their behaviour may not be since a common dimension of a physical quantity does not imply a common value.
Proposition 4.1
The dimensional groups \(\mathcal {G}_{\beta }\) and \(\mathcal {G}_{ps}\) coincide.
Proof 4.1
The proof follows immediately from the identities \(L_{\beta }=L_{ps}\), \(T_{\beta }=T_{ps}\) and \(M_{\beta }=M_{ps}\). \(\square \)
Note that although the proposition here is limited to mechanics it is fairly evident that it extends to other physical theories and indeed to non-physical theories also. Effectively, no matter what the theory, the space \(\Omega _{\beta }\) is constructed to ensure the groups \(\mathcal {G}_{\beta }\) and \(\mathcal {G}_{ps}\) coincide.
4.1 Curves and tangent vectors of the dimensional group
It is appreciated that dimensional groups are Lie groups [45] and the ability to scale dimensions is invoked without resorting to previous justifications of a change in units. The dimensional groups \(\mathcal {G}_{\beta }, \mathcal {G}_{ps}\), and \(\mathcal {G}_{ts}\) are finite-dimensional abelian Lie groups since the multiplication is commutative and as such are isomorphic to \((\mathbb {R}^n,+)\) [46]. The isomorphism is confirmed by the map \(\mathcal {G}_\beta \rightarrow (\mathbb {R}^n,+)\) in the form \(\phi ([\mathbf {\Psi }_\beta ]=[a_\beta ^1]^{r_1} \cdots [a_\beta ^n]^{r_n})=(r_1,...,r_n)_\beta \), taking \([a_\beta ^i]\) to be independent dimensions generating all elements in the group (e,g., \(L_\beta \), \(T_\beta \), and \(M_\beta \)). The group \(\mathcal {G}_\beta \) is a Lie group with its associated Lie algebra or tangent space \(\mathcal {T}_\beta \) whose elements are representable by tangent vectors, evidently portrayed by the identity \(\phi ([\mathbf {\Psi }_\beta ])=r_1\mathbf {e_\beta ^1}+\cdots +r_n\mathbf {e_\beta ^n}\) with \(\{\mathbf {e_\beta ^i}\}\) the canonical base of \((\mathbb {R}^n,+)\). The distinction made above between the dimensions in the different spaces is reflected in the canonical base. Consequently, the construction of \(\mathcal {G_\beta }\) immediately defines a parametrised curve \(\chi _{[\mathbf {\Psi }_{\beta }]} (\beta )\) on setting \([a_{ts}^i]=\alpha ^{[a^i]}(\beta )[a_{\beta }^i]\) (e.g., \(L_{ts} =\beta L_{\beta }\), \(T_{ts} =g(\beta ) T_{\beta }, M_{ts}=\alpha ^{\rho }(\beta )M_{\beta }\)) for smooth non-zero functions \(\alpha ^{[a^i]}(\beta )\), given by
Observe that the curve is a consequence of the distinction being made between the base dimensions in \(\mathcal {G}_{ts}\) and \(\mathcal {G}_{\beta }\) with the vectors in the tangent spaces being related by \(\textbf{e}_{ts}^i=\alpha ^{[a^i]}(\beta )\textbf{e}_{\beta }^i\).
4.2 Lie transport of scaled physical quantities
With the establishment that scaling dimensions can be viewed as a scaling trajectory of the dimensional group attention now turns to the transport of physical quantities. The first concern is the correct identification of the physical quantities to be transported. The finite similitude theory confirmed for mechanics that these are \(\textbf{v}_{\beta }^*=g \beta ^{-1} \textbf{v}_{ts}^*\), \(\rho _{\beta }=\alpha ^\rho _0\beta ^3\rho _{ts}\), \(\textbf{v}_{\beta }=\beta ^{-1}g\textbf{v}_{ts}\), \(\textbf{u}_{\beta }=\beta ^{-1}\textbf{u}_{ts}\), \(\mathbf {\sigma }_{\beta }=\alpha ^\rho _0g^2\beta \mathbf {\sigma }_{ts}\), and \( \textbf{b}^v_{\beta }=g^2\beta ^{-1}\textbf{b}^v_{ts}\), which are represented here by the quantity \(\mathbf {\Psi }_\beta \). These quantities ensure the correct form of the transport equation on \(\Omega _{\beta }\), but as revealed in Sect. 3.2 also ensure \(\Omega _{ps}\) and \(\Omega _{\beta }\) coincide dimensionally. The objective here is to seek invariants and in general terms examine how \(\mathbf {\Psi }_{\beta }\) varies with \(\beta \), where \(\beta \) scales the principal quantity (e.g., space, capital, information quantity). Note that \([\mathbf {\Psi }_{\beta }]\) is a fixed point in \(\mathcal {G}_\beta \) being characterised by its dependency on the base dimensions (e.g., scalars \(r_1\), \(r_2\), and \(r_3\) for dimensions \([a_1]=[L_\beta ]=[L_{ps}],[a_2]=[T_\beta ]=[T_{ps}]\), and \([a_3]=[M_\beta ]=[M_{ps}]\)). The maps \(\mathbb {R}^+\rightarrow \mathcal {P}_\beta \rightarrow \mathcal {G}_\beta \) are of interest, where \(\mathcal {P}_\beta \) is a (physical) manifold for properties in \(\Omega _{\beta }\), where in the first instance \(\beta \mapsto \mathbf {\Psi }_{\beta }\mapsto \alpha _0^\Psi [\mathbf {\Psi }_\beta ]\) is the focus. More generally, curves of interest are \(\beta \mapsto \alpha _0^\Psi [\mathbf {\Psi }_\beta ]\), \(\beta \mapsto \alpha _1^\Psi [\tfrac{d }{{\text {d}} \beta }(\alpha _0^\Psi \mathbf {\Psi }_{\beta })]\), \(\beta \mapsto \alpha _2^\Psi [\tfrac{d }{{\text {d}} \beta }(\alpha _1^\Psi \tfrac{d }{{\text {d}} \beta }(\alpha _0^\Psi \mathbf {\Psi }_{\beta }))]\) and so on. These curves can be made independent of \(\beta \) (for this is the objective) if they respectively satisfy the identities \(\tfrac{d }{{\text {d}} \beta }(\alpha _0^\Psi \mathbf {\Psi }_{\beta }) \equiv \textbf{0}\), \(\tfrac{d }{{\text {d}} \beta }(\alpha _1^\Psi \tfrac{d }{{\text {d}} \beta }(\alpha _0^\Psi \mathbf {\Psi }_{\beta }) )\equiv \textbf{0}\), \(\tfrac{d }{{\text {d}} \beta }(\alpha _2^\Psi \tfrac{d }{{\text {d}} \beta }(\alpha _1^\Psi \tfrac{d }{{\text {d}} \beta }(\alpha _0^\Psi \mathbf {\Psi }_{\beta })))\equiv \textbf{0}\) and so on. Consequently, for \(\alpha _0^\Psi =1\), invariance with respect to \(\beta \) in the transport of \(\alpha _0^\Psi [\mathbf {\Psi }_\beta ]\) coincides with the zeroth-order condition \(\tfrac{d }{{\text {d}} \beta }\mathbf {\Psi }_{\beta } \equiv \textbf{0}\) since integration immediately returns \(\mathbf {\Psi }_{\beta }=\mathbf {\Psi }_{1}\) and if true for all quantities, then scaled behaviour matches full-scale behaviour. If however, \(\tfrac{d }{{\text {d}} \beta }\mathbf {\Psi }_{\beta } \not \equiv \textbf{0}\) for one or more of the physical quantities, then the next curve is obtained by scaling the derivative \(\tfrac{d }{{\text {d}} \beta }\mathbf {\Psi }_{\beta }\), i.e., form \(\alpha _1^\Psi \tfrac{d }{{\text {d}} \beta }\mathbf {\Psi }_{\beta }\) and consider \(\beta \mapsto \alpha _1^\Psi [\tfrac{d }{{\text {d}} \beta }\mathbf {\Psi }_{\beta }]\). The invariant of interest is \(\tfrac{d }{{\text {d}} \beta }(\alpha _1^\Psi \tfrac{d }{{\text {d}} \beta }\mathbf {\Psi }_{\beta })\equiv \textbf{0}\), which is, of course, the first-order finite similitude rule and, using the mean-value theorem and two trial scales, integrates exactly to give
which is identical in form to those in the second column of Table 1.
Clearly, the process can be continued considering next the path given by \(\beta \mapsto \alpha _2^\Psi [\tfrac{d }{{\text {d}} \beta }(\alpha _1^\Psi \tfrac{d }{{\text {d}} \beta }\mathbf {\Psi }_{\beta })]\) and the invariance \(\tfrac{d }{{\text {d}} \beta }(\alpha _2^\Psi \tfrac{d }{{\text {d}} \beta }(\alpha _1^\Psi \tfrac{d }{{\text {d}} \beta }\mathbf {\Psi }_{\beta }))\equiv \textbf{0}\), which integrates exactly to give
which is identical in form to those in the third column of Table 1.
It is worth noting that there is no barrier to the R-parameters in Eq. (21) and Eq. (22) being negative and in particular \(R_1=-1\) reduces Eq. (21) to \(\mathbf {\Psi }_{1}=\mathbf {\Psi }_{\beta _2}\), i.e., a zeroth-order relationship between processes in \(\Omega _{ps}\) and \(\Omega _{\beta _2}\). Examination of Eq. (10) indicates that \(R_1=-1\) would only be possible if \(\alpha _1^\Psi \) takes up both negative and positive values. This shows that the scaling trajectory of \(\beta \mapsto \alpha _1^\Psi [\tfrac{d }{{\text {d}} \beta }(\mathbf {\Psi }_{\beta })]\) can involve the multiplication of the dimension by negative real numbers. Figure 4 illustrates the connection between the finite similitude approach and the dimensional approach presented here. The curve \(\chi _{[\mathbf {\Psi }_{\beta }]}(\beta )\) depicted in the figure is a projection due to the change in base dimensions described above.
Fig. 4
The figure shows the finite-similitude transport approach codified by the dimensional approach constituting a Lie group \(\mathcal {G}_\beta \), and curves that characterise the scaling relationships. The dimensions of quantities in \(\Omega _{\beta }\) are represented as elements of \(\mathcal {G_\beta }\), where the arrowed black line represents scaling trajectories parametrised by \(\beta \). From this geometrical perspective it is immediate to observe that all similitude rules correspond to fixed points along the projected curves \(\alpha ^\Psi _1[\phi (\tfrac{d }{{\text {d}} \beta }\mathbf {\Psi }_{\beta })] \), \(\alpha ^\Psi _2[\phi (\tfrac{d }{{\text {d}} \beta }(\alpha _1^\Psi \tfrac{d }{{\text {d}} \beta }\mathbf {\Psi }_{\beta }))]\) and so on, as a consequence of their invariance with respect to \(\beta \)
×
Proposition 4.2
The finite similitude scaling theory for physical systems is equivalent to an extension of dimensional analysis on matching the groups \(\mathcal {G}_{ps}\) and \(\mathcal {G}_{\beta }\) and adopting identical similitude identities.
Proof 4.2
The proof follows immediately from Prop. 4.1 and the discussion in this section. \(\square \)
It is clear that the finite similitude approach and the extended form of dimensional analysis are ultimately seeking scale invariances under the action of a single scaling parameter \(\beta \). This can be contrasted with Lie symmetries that similarly arise out of 1-parameter group transformations of solutions to differential equations. The focus here however is not solutions to differential equations but on extending the definition of scale symmetries through new forms of similitude rules whose solution is the governing equations (and by default their solution) as presented in \(\Omega _{ps}\).
5 Scaling finance
In view of the equivalence between finite similitude and the extended dimensional analysis from the previous section it is of interest to explore the reach of the extended approach. Dimensional analysis applies to all quantitative-based theories and is not tied to physical transport equations. The Black-Scholes equation for option pricing is considered in this section since although outside physics it nevertheless has a connection in that the equation is identical in form to that describing diffusive heat transfer. The equation takes the form of a backward parabolic partial differential equation, i.e.,
where \(v_{ts}\) is the value of an option (either call or put), \(s_{ts}\) is the asset (stock) price \(\sigma _{ts}\) is the volatility of the underlying asset, and \(r_{ts}\) is the interest rate, and where typically both \(\sigma _{ts}\) and \(r_{ts}\) may be assumed to be reasonably constant over the lifetime of the contract, and where the boundary and "final" conditions are set as \(v_{ts}(a_{ts},t_{ts})=v^a_{ts}(t_{ts})\), \(v_{ts}(b_{ts},t_{ts})=v^b_{ts}(t_{ts})\) and \(v_{ts}(s_{ts},e_{ts})=v^e_{ts}(s_{ts})\), where \(e_{ts}\) is the contract expiry.
5.1 Black-Scholes on \(\Omega _{\beta }\)
Although the Black-Scholes can be written in transport form the dimensional equivalence of \(\mathcal {G}_\beta \) and \(\mathcal {G}_{ps}\) means that it is relatively straightforward to express Eq. (23) on \(\Omega _\beta \). It is simply a matter of defining the dimensional relationships for value \(S_\beta =\beta ^{-1}S_{ts}=S_{ps}\) and time \(T_\beta =g^{-1}T_{ts}=T_{ps}\). It immediately follows that \(s_\beta =\beta ^{-1}s_{ts}\), \(t_\beta =g^{-1}t_{ts}\), \(v_\beta =\beta ^{-1}v_{ts}\), \(\sigma _\beta =g^{1/2}\sigma _{ts}\) and \(r_\beta =gr_{ts}\) and consequently Eq. (23) projects to
confirming the form of the Black-Scholes equation on \(\Omega _\beta \).
Note additionally that the underpinning stoichiometric ordinary differential equation \(\frac{d s_{ts}}{s_{ts}}=\sigma _{ts}dx_{ts}+\mu _{ts}dt_{ts}\) for Eq. (23) has the equivalent equation \(\frac{d s_{\beta }}{s_{\beta }}=\sigma _{\beta }dx_{\beta }+\mu _{\beta }dt_{\beta }\) on \(\Omega _\beta \). Here \(dx_{ts}\) is assumed to follow a random Weiner process with zero mean and variance \(d t_{ts}\) (i.e., \(dx_{ts}=\phi \sqrt{dt_{ts}}\) with \(\phi \sim {\textbf {Normal}}(0,1)\)), and where \(\mu _{ts}\) is the mean. The quantities are related to associated quantities on \(\Omega _{\beta }\) by \(dx_{\beta }=g^{-1/2}dx_{ts}\) and \(\mu _{\beta }=g\mu _{ts}\), which follows immediately from dimensional considerations. With the formulation of Eq. (24) it is now possible to consider what similitude rules (if any) apply, which is initiated by the setting of the zeroth-order conditions \(\frac{d}{{\text {d}} \beta } s_\beta \equiv 0\) and \(\frac{d}{{\text {d}} \beta } t_\beta \equiv 0\). The simplest assumption is that the zeroth-order identity applies which means that Eq. (24) simply vanishes on differentiation with respect to \(\beta \), which is a result that implies (not too unexpectedly) there is no size effect in a single asset. A more realistic market scaling is considered in the section but prior to that is of interest to examine the situation where the interest rate \(r_\beta \) changes with scale, where improved terms are provided for larger asset values. In view of the product terms present in Eq. (24) is is of interest to examine a second-order solution obtained on limiting \(r_\beta \) and \(v_\beta \) to first order. This means that (24) satisfies
where \(\alpha _1\) and \(\alpha _2\) are smooth functions of \(\beta \).
Attempting to solve this equation analytically poses some difficulty as it contains unknown functions \(\alpha _1\) and \(\alpha _2\). Although, as mentioned above, symmetry methods may be utilised to find exact solutions (to the Black-Scholes [47]), the focus here is on recovering behaviour in \(\Omega _{ps}\), which is achieved immediately on application of similitude rules in Sect. 2.2. These utilise the mean-value theorem and free scaling parameters, and provide expressions that are easily and readily implemented in practical applications as follows:
which are solutions to the identities \(\tfrac{d}{{\text {d}} \beta }(\alpha _1 \tfrac{d}{{\text {d}} \beta }r_\beta )\equiv 0\) and \(\tfrac{d}{{\text {d}} \beta }(\alpha _1 \tfrac{d}{{\text {d}} \beta }v_\beta )\equiv 0\) and inspection of Eq. (24) confirms the requirement \(\alpha _1=\alpha _2\) and the assumption \(\tfrac{d}{{\text {d}} \beta }\sigma _\beta \equiv 0\).
It is evident that although the similitude theory has been shown to be applicable outside of physics the Black-Scholes example offers little scope for scale effects. A portfolio of higher complexity is required and it transpires that there is published evidence of second order behaviour in the financial markets.
5.2 Scale effects in the financial markets
The extension of finite similitude, utilising its dimensional implementation in quantitative-based theories, allows for the consideration of case studies in which scaling effects appear to be present. One of these pertains to asset and security pricing in financial mathematics, more specifically, the aptly named "size effect" first encountered by Banz in reference [48] when analysing historical data relating to securities. He postulated that the standard model, attributed to Black-Scholes [49], is misspecified since there appeared to be a mismatch in the residuals which were expected to be zero. It was observed that the expected returns from portfolios of securities with smaller market proportions were greater than those associated with larger market proportions. This sparked considerable debate in the literature [50‐52] as to whether the effect existed at all or was an artefact of the data. The purpose of this analysis is not to definitively determine the existence of the "size effect", but rather provide a mathematical model to study such effects in quantitative-based theories; not tied to physical transport equations. It is explicitly stated in reference [48] that there exists no theoretical foundation for such an effect and it is plausible that size is just a proxy for other unknown factors correlated with size. Nonetheless, it is of interest to examine how the finite-similitude approach can be applied, since (as established in this work) it provides the means to describe scale and size effects in quantitative theories.
A generalised asset pricing model can be stated as a linear relationship in the form
where \(E[R_A(\beta ,t)]\) is the expected return of security A, \(\sigma _A\) is the risk of security A, \(\gamma _0\) is the expected return on a zero-sigma portfolio, \(\gamma _1\) is the expected market premium, \(\phi _A\) the value of security A, \(\phi _m\) the average market value and \(\gamma _2\) is a constant measuring the contribution of the market proportion to the expected return.
If no relationship exists between the market value and the expected return (\(\gamma _2=0\)), then Eq. (27) reduces to the Black version of the capital pricing model [49]. The parameters are estimated using historical or empirical data (where \(\phi _A\) and \(\phi _m\) are defined as market proportion); methods include pooled cross-sectional and time series regressions to estimate \(\gamma _i\). To generate minimum variance portfolios with mean returns \(\gamma _i\), the following constraints are imposed
where \(w_j\) are the portfolio proportions for each asset j; with the minimum variance condition \(\sum _j w_j=1\).
Reference [48] utilises historical data for all common stocks quoted on the NYSE for at least five years between 1926 and 1975. A plethora of statistical considerations for the estimation of these parameters are discussed at length in reference [48], one of which is the manner in which securities are assigned to one of twenty-five portfolios on the basis of market value/proportion. Firstly, they are assigned to one of five on the basis of market proportion and, consequently, to one of five on the basis of their stochastic term \(\sigma \). Once the parameters have been statistically derived, the analysis of residuals of the twenty-five portfolios is undertaken in the form
The approximated values of this calculation are presented in Table 2 (Appendix A) which are used in reference [48] to produce the values of the 25 portfolios plotted in Fig. 5; where the residual return is plotted against the market proportion or security "size" value. Statistically, the expected return should be zero, however, the plot shows portfolios with smaller market proportions having a greater return than their larger counterparts.
Fig. 5
Figure with the values from Table 2 from reference [48] where the numbers 1,..., 5 represent the residual return (calculated using eq. (29)) for the five portfolios in each size group (i.e. 1 corresponds to portfolio with largest sigma and 5 the smallest sigma). The orange markers "x" represent the mean centre of two adjacent size groups (10 data values in total) which are used to determine the length-scale parameters \(\beta _0,...,\beta _3\). The intervals that correspond to each of the four scales are illustrated at the bottom of the plot
×
It is assumed that the size effect is independent of the statistical parameters previously estimated and, thus, a consequence of an extra term \(\epsilon (\beta ,t)\). In practical applications this term can be regarded as the "fees" associated to the return of a security dependent on its size (taxes, management fees, amongst others). The equation analysed is as follows:
which is similar to Eq. (28) apart from the addition of \(\epsilon _P(\beta ,t)\).
Second-order similitude is considered to apply as it provides three degrees of freedom in the form of free parameters \(R_{1,1}\), \(R_{1,2}\) and \(R_2\), which are calculated using the statistical data to fit the relationship that more closely approximates the results presented in Fig. 5 by minimising the error in the similitude relationships. Assuming second-order relationships the residual returns conform to
where the scales fulfil the condition \(\beta _3<\beta _2<\beta _1<\beta _0\).
To simplify the application of the second-order approach and set the length-scale parameters \(\beta _0,...,\beta _3\), the mean value of the market proportion between the portfolios in each pair of sizes is calculated and corresponds to the horizontal value of the orange markers "x" in Fig. 5. Utilising these scales the objective is minimising the function
where \(\overline{\epsilon _{i}}=\epsilon _P(\beta _i)\beta _i^{-1}\) with \(i=0,1,2,3\) and \(T_1=R_{1,1}, T_2=R_2R_{1,1}, T_3=R_2,R_{1,1}R_{1,2}\), and where \(T_{add}=T_1+T_2\).
The optimised parameters are readily obtained on minimisation of \(h(T_{add},T_3)\), which provides \(T_{add}=-0.258247\) and \(T_3=0.0514705\), and on arbitrarily setting \(T_2=1\), the final result is \(T_1=-1.258247\), \(T_2=1\), and \(T_3=0.0514705\). Consequently, the second-order relationship returned is
which provides a practical application of a second-order similitude relationship outside the field of physics.
The size effect has been accounted for and, being a second-order similitude rule, three scales can be used to predict or approximate a specific market proportion residual return relying solely on the size-effect relationships. In this case, it is possible to use data from portfolios in three-scale (\(\beta _i\)) intervals as shown in Fig. 5 to predict the values for the remaining interval. Since the application has been simplified for demonstration purposes the accuracy of the approximation relies heavily on the proximity of the selected data values to the means (orange markers "x") used to derive parameters \(T_i\).
6 Conclusion
The focus of this paper is on the extension of the finite similitude theory to all quantitative-based sciences, which is achieved through an extension to dimensional analysis. The recently introduced scaling space \(\Omega _{\beta }\) is shown to be the key to this extension and despite being a mere projection of experimental-space physics can nevertheless be viewed as a real physical space, in mathematical terms. This observation and confirmation that the full-scale physical space is dimensionally equivalent provides the means to take finite similitude beyond the physical sciences. It is shown in the paper that similitude rules, as scale-invariance relationships, can be applied directly to any formulation within the context of scaling approaches and all that is required is the establishment of quantities on \(\Omega _{\beta }\); emphasising its importance. The following conclusions can be drawn from the work presented in the paper:
The theory of finite similitude has been further developed with the formation of the space \(\Omega _{\beta }\) as a new metaphysical space for any \(\beta \), with dimensional and physical equivalence to experimental spaces.
Dimensional analysis is extended in the work by abandoning the notion of dimensionless physics and identifying a critical distinction between dimensions in different spaces, thus formalising scaling relationships across spaces.
Similitude identities have been defined in terms of invariant relationships (matching those of the finite similitude theory), on consideration of scaling paths projected onto the dimensional Lie group \(\mathcal {G}_\beta \), providing an extended dimensional framework to scale quantitative-based systems.
More specifically, from the financial study it has been show that:
The new extended scaling theory overcomes the weaknesses of previous scaled-dimensional approaches in finance and is able to accommodate size effects in interest rates, being applicable to the Black-Scholes equation.
The theory provides a practical framework to account for size effects in financial markets based on the statistical data reported in reference [48]. The approach results in explicit second-order relationships that can be utilised to estimate the return of assets/portfolios in a specific market proportion using data from three "trial" scales or proportions; relying solely on size-effect relationships and the CAPM.
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