The scaling of stochastic physical systems with applications in electrodynamics

The starting point for finite similitude is the imagined concept of space scaling, where space itself is either expanded or contracted isotropically as dictated by the positive real parameter \(\beta \). A relationship is established between inertial coordinates in the physical space \(x^i_{ps}\) (where the full-size system resides) to coordinates in the trial space \(x^i_{ts}\) (where the scaled system sits). This relationship is expressed in differential terms as \(d{\textbf{x}}_{ts} = \beta d{\textbf{x}}_{ps}\), describing isotropic scaling in the case of orthonormal inertial frames. Spacetime manifolds are established in the two spaces and are connected through the two differential identities \(\mathrm{{d}}t_{ts} =g \mathrm{{d}}t_{ps}\) and \(\mathrm{{d}}{\textbf{x}}_{ts} = \beta \mathrm{{d}}{\textbf{x}}_{ps}\), where g is a positive function of \(\beta \). Since each coordinate and instance in time are in no way special, the spacetime manifolds are assumed flat with g being both temporally and spatially invariant. This arrangement ensures that space contraction is identified by \(\beta < 1\), with no change for \(\beta =1\), and expansion identified with \(\beta >1\). Isotropic scaling tends to be the most useful form of scaling, as it applies universally to all things and situations [23, 31] but nonetheless anisotropic space scaling is feasible but not considered further here (see [39, 40] for discussion and applications).

The genesis of the finite-similitude theory is space through its connection to physical systems by representations in transport form. These forms of governing equation permit the response of a system to be described on a moving region of space (a.k.a. a control volume), and consequently, it becomes transparent how space scaling impacts through its effect on the size of a control volume. The motions of control volumes in physical and trial spaces are connected with details presented in references [44, 45, 47]. The general thrust is to assume their exists a relationship (a similitude rule) between velocity fields \({\textbf{v}}_{ps}^*\) and \({\textbf{v}}_{ts}^*\), which dictate the motions of control volumes in the physical and trial spaces, respectively. To make precise mathematically, the transport of control volumes \(\Omega ^*_{ts}\) and \(\Omega ^*_{ps}\), the following temporal derivatives \(\tfrac{D^*}{D^* t_{ts}}=\tfrac{\partial }{\partial t_{ts}} \big |_{\mathbf {\chi }_{ts}}\) and \(\tfrac{D^*}{D^* t_{ps}}=\tfrac{\partial }{\partial t_{ps}} \big |_{\mathbf {\chi }_{ps}}\) are defined, where \(\mathbf {\chi }_{ts}\) and \(\mathbf {\chi }_{ps}\) are coordinate points in reference control volumes \(\Omega ^{*ref}_{ts}\) and \(\Omega ^{*ref}_{ps}\), respectively. The general idea is presented in Fig. 1a , showing how moving control volumes \(\Omega ^*_{ts}\) and \(\Omega ^*_{ps}\) (along with reference control volumes \(\Omega ^{*ref}_{ts}\) and \(\Omega ^{*ref}_{ps}\)) are related. In mathematical terms, the motion of the moving control volumes is 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 control-volume points. The arrangement presented in Fig. 1a encapsulates the initial approach to finite similitude but lacks the scope to distinguish between predicted similitude-rule behaviour and actual behaviour taking place in the physical space. This was resolved in more recent work with the introduction of the scaling space \(\Omega _{\beta }\) [31].

This space is formulated through the introduction of the subscript \(\beta \) notation and the projection of physics from the trial space \(\Omega _{ts}\). This is initiated by scaling the trial spacetime coordinates to form \(d{\textbf{x}}_{\beta } = \beta ^{-1} d{\textbf{x}}_{ts}\) and \(d{t}_{\beta } = g^{-1} d{t}_{ts}\), along with the control-volume velocity \({\textbf{v}}_{ts}^*\) to form \({\textbf{v}}_{\beta }^*=g \beta ^{-1} {\textbf{v}}_{ts}^*\). The subscript \(\beta \) infers \(\beta -\)dependence, and similitude rules (if satisfied) dictate what changes are possible. The simplest similitude rule (the zeroth-order rule) is no change with \(\beta \) and for these quantities returns \(\tfrac{d}{d\beta }d{\textbf{x}}_{\beta }\equiv {\textbf{0}}\), \(\tfrac{d}{d\beta }dt_{\beta }\equiv {0}\) and \(\tfrac{d}{d\beta }{\textbf{v}}_{\beta }^*\equiv {\textbf{0}}\), which on integration give \(d{\textbf{x}}_{\beta }=\mathrm{{d}}{\textbf{x}}_1=d{\textbf{x}}_{ps}\), \(dt_{\beta }=dt_1=dt_{ps}\), and \({\textbf{v}}_{\beta }^*={\textbf{v}}_1^*={\textbf{v}}_{ps}^*\), respectively. Note that the identity symbol "\(\equiv \)" here signifies that the identity holds true \(\forall \beta >0\). The similitude rules connect the scaling space \(\Omega _\beta \) to the physical space \(\Omega _{ps}\), and it is evident that applying the zeroth-order rule to \(d{\textbf{x}}_{\beta }\), \(dt_{\beta }\), and \({\textbf{v}}_{\beta }^*\) ensures that in the two spaces, timescales, dimensions, and moving control-volume motions are identical. Presented in Fig. 1b is a schematic of the concepts involved with relationships between the control volumes in the spaces \(\Omega _{ps}\), \(\Omega _{\beta }\), and \(\Omega _{ts}\). The dashed line connecting control volumes \(\Omega _{ps}^*\) and \(\Omega _{\beta }^*\) indicates that a similitude rule (here the zeroth-order rule) has been applied to relate their motions. The velocity field \({\textbf{v}}_{\beta }^*\) satisfies the differential identity \(\mathrm{{d}}{\textbf{x}}_{\beta }^*={\textbf{v}}_{\beta }^* \mathrm{{d}}t_{\beta }\), and consequently, on substitution of \(\mathrm{{d}}{t}_{\beta } = g^{-1} \mathrm{{d}}{t}_{ts}\) and \({\textbf{v}}_{\beta }^*=g \beta ^{-1} {\textbf{v}}_{ts}^*\) gives the differential identity \(\mathrm{{d}}{\textbf{x}}_{\beta }^*= \beta ^{-1}{\textbf{v}}_{ts}^*\mathrm{{d}}t_{ts}=\beta ^{-1}\mathrm{{d}}{\textbf{x}}_{ts}^*\), which can be contrasted against the space-scaling map \( \mathrm{{d}}{\textbf{x}}_{\beta }=\beta ^{-1}\mathrm{{d}}{\textbf{x}}_{ts}\). The identity leads immediately to the geometric measure relationships \(\mathrm{{d}}{V}^*_{\beta } = \beta ^{-3} \mathrm{{d}}{V}^*_{ts}\) and \(\mathrm{{d}}\mathbf {\Gamma }^*_{\beta } = \beta ^{-2} \mathrm{{d}}\mathbf {\Gamma }^*_{ts}\) (for volume and area, respectively), where \(\mathrm{{d}}\mathbf {\Gamma }^*_{\beta } = {\textbf{n}}^*_{\beta }\mathrm{{d}}{\Gamma }^*_{\beta }\), and where \({\textbf{n}}^*_{\beta }\) is a unit vector.

All the pieces are now in place to introduce physics to the space \(\Omega _\beta \) through transport equations on the trial space \(\Omega _{ts}\), which takes the generic form,where \(\mathbf {\Psi }_{ts}\) is the principal field of interest, \(\rho _{ts}\) is material density, \({\textbf{v}}_{ts}\) is material velocity, and \({\textbf{b}}_{ts}^\psi \) and \({\textbf{J}}_{ts}^\psi \) are source and flux terms, respectively.

Substitution of the geometric measures \(\mathrm{{d}}{V}^*_{ts} = \beta ^3 \mathrm{{d}}{V}^*_{\beta }\), \(\mathrm{{d}}\mathbf {\Gamma }^*_{ts} = \beta ^2 \mathrm{{d}}\mathbf {\Gamma }^*_{\beta }\), and temporal relationship \(\mathrm{{d}}{t}_{ts} = g \mathrm{{d}}{t}_{\beta }\) into Eq. (1) gives on multiplication throughout by \(\alpha _0^\psi g\) (with \(\alpha _0^\psi \) a smooth function of \(\beta \)),where \(\rho _{\beta } \mathbf {\Psi }_{\beta }=\alpha _0^\psi \beta ^3 \rho _{ts} \mathbf {\Psi }_{ts}\), \({\textbf{v}}_{\beta }^*=g \beta ^{-1} {\textbf{v}}_{ts}^*\), \({\textbf{v}}_{\beta }=g \beta ^{-1} {\textbf{v}}_{ts}\), \({\textbf{J}}_{\beta }^\psi =\alpha _{0}^\psi \beta ^2 g {\textbf{J}}_{ts}^\psi \), and \({\textbf{b}}_{\beta }^\psi =\alpha _{0}^\psi \beta ^3 g {\textbf{b}}_{ts}^\psi \), and where this equation captures the physics of the trial space without approximation and is central to the finite-similitude theory being a projection of Eq. (1) on \(\Omega _{\beta }\).

Its validity ranges for all values of \(\beta \) (assuming Eq. (1) is valid) and it returns the transport equation in the physical space for \(\beta =1\), with \(\alpha _{0}^\psi (1)=1\), \(g(1)=1\), and \( \rho _{1} \mathbf {\Psi }_{1}=\rho _{ps} \mathbf {\Psi }_{ps}\), \({\textbf{v}}_{1}^*={\textbf{v}}_{ps}^*\), \({\textbf{v}}_{1}={\textbf{v}}_{ps}\), \({\textbf{J}}_{1}^\psi = {\textbf{J}}_{ps}^\psi \), and \( {\textbf{b}}_{1}^\psi ={\textbf{b}}_{ps}^\psi \). A critical point to note is that Eq. (2) is exactly of the form expected of a transport equation in a real physical space. This immediately signifies that physics in the scaling space \(\Omega _\beta \) can be treated in a traditional manner without recourse to transport forms as required, and consequently, stochastic differential equations can in principle be investigated.

The link between dimensional analysis and finite similitude was established in Ref. [31], where it is confirmed that the dimensional groups \({\mathcal {G}}_\beta \) and \({\mathcal {G}}_{ps}\) coincide. It is well appreciated that dimensions form a group called the dimensional group [48], but in Ref. [31], a distinction is made between groups of dimensions in the spaces \(\Omega _{ps}\), \(\Omega _{ts}\), and \(\Omega _\beta \) denoted as \({\mathcal {G}}_{ps}\), \({\mathcal {G}}_{ts}\), and \({\mathcal {G}}_{\beta }\), respectively. Although the dimension of a physical quantity can be identified with a scalar function relating different units [15], this definition is insufficient for the purposes of distinguishing dimensions in different spaces. A definition more suited is the identification of dimensions with properties that are measurable in the broadest sense that they can be quantified. A common system of units is assumed since changing units in the different physical spacetime manifolds is of little concern. Similarly, it is common practice to identify pure dimensions (in physics) for space, time, mass, temperature, and charge with the uppercase letters L, T, M, \(\Theta \), and Q, respectively. Likewise, square bracket notation is often adopted to identify the dimension of a quantity, i.e. \(T=[t]\) and \(L=[{\textbf{x}}]\), for time and space, respectively. With the view that dimensions are distinct in each space, pure dimensions are labelled with the appropriate subscript, e.g. \(L_{ts}=[{\textbf{x}}_{ts}]\) & \(T_{ts}=[t_{ts}]\), \(L_{ps}=[{\textbf{x}}_{ps}]\) & \(T_{ps}=[t_{ps}]\), and \(L_{\beta }=[{\textbf{x}}_{\beta }]\) & \(T_{\beta }=[t_{\beta }]\). In the field of mechanics pure members of the group \({\mathcal {G}}_{ts}\) are the identity \(1_{ts}\), time \(T_{ts}=[t_{ts}]\), space \(L_{ts}=[{\textbf{x}}_{ts}]\), mass \(M_{ts}=[m_{ts}]\) and similarly for \({\mathcal {G}}_{ps}\) with a change in subscript ("ps" replaces "ts"). Members of \({\mathcal {G}}_{\beta }\) however are returned on scaling those members in \({\mathcal {G}}_{ts}\), which is of course permissible since dimensional groups are also Lie algebras [48]. The pertinent scaling relationships are \(\beta ^{-1}L_{ts}=L_{\beta }\), \(g^{-1}T_{ts}=T_{\beta }\), and \(\alpha _{0}^\rho M_{ts}=M_{\beta }\), where \(\alpha _{0}^\rho \) is the scalar applied to the continuity equation in transport form (more on this below). The dimensional groups \({\mathcal {G}}_{ps}\) and \({\mathcal {G}}_{\beta }\) coincide on setting \(L_{ps}=\beta ^{-1}L_{ts}\), \(T_{ps}=g^{-1}T_{ts}\) and \(M_{ps}=\alpha _{0}^{\rho } M_{ts}\). Note however (for mechanics), despite spaces \(\Omega _{\beta }\) and \(\Omega _{ps}\) being identical dimensionally, physical responses may differ significantly since common dimensions does not imply common values. It is revealing to contrast this approach with the direct scaling of transport equations, which requires no recourse to dimensional analysis. The scaling space \(\Omega _\beta \) permits transport equations pertinent to mechanics to be immediately presented in standard form. The relevant equations are the non-conserved movement equation (introduced in Ref. [49]) and the conserved equations for volume, mass, and momentum. The advantage offered by transport equations is that they are conceptual and require no detailing of the specific problem and in this respect is similar to the approach of dimensional analysis. The transport equations of interest contain fields of concern, which are displacement \({\textbf{u}}_\beta \), control-volume velocity \({\textbf{v}}_{\beta }^*\), material density \(\rho _{\beta }\), material velocity \({\textbf{v}}_{\beta }\), Cauchy stress \(\varvec{\sigma }_{\beta }\), and body force density \({\textbf{b}}_\beta \). Explicitly stated the transport equations in \(\Omega _{\beta }\) are

where \({\textbf{v}}_{\beta }^*=g \beta ^{-1} {\textbf{v}}_{ts}^*\), \(\rho _{\beta }=\alpha ^\rho _0\beta ^3\rho _{ts}\), \({\textbf{v}}_{\beta }=\beta ^{-1}g{\textbf{v}}_{ts}\), \({\textbf{u}}_{\beta }=\beta ^{-1}{\textbf{u}}_{ts}\), \(\varvec{\sigma }_{\beta }=\alpha ^v_0g\beta ^2\varvec{\sigma }_{ts}\), \( {\textbf{b}}_{\beta }=\alpha ^v_0g\beta ^3{\textbf{b}}_{ts}\), and where the setting \(\alpha _0^1=\beta ^{-3}\), \(\alpha ^v_0=g\beta ^{-1}\alpha ^\rho _0\), and \(\alpha ^u_0=\beta ^{-1}\alpha ^\rho _0\) are necessary to achieve the expected form.

It is apparent that the relationships \({\textbf{v}}_{\beta }^*=g \beta ^{-1} {\textbf{v}}_{ts}^*\), \(\rho _{\beta }=\alpha ^\rho _0\beta ^3\rho _{ts}\), \({\textbf{v}}_{\beta }=\beta ^{-1}g{\textbf{v}}_{ts}\), \({\textbf{u}}_{\beta }=\beta ^{-1}{\textbf{u}}_{ts}\), \(\varvec{\sigma }_{\beta }=\alpha ^v_0g\beta ^2\varvec{\sigma }_{ts}\), and \({\textbf{b}}_{\beta }=\alpha ^v_0g\beta ^2{\textbf{b}}_{ts}\), along with the identity \(\alpha ^v_0=g\beta ^{-1}\alpha ^\rho _0\), can be deduced from dimensional considerations alone. To see this, note first that the kinematic terms provide \([{\textbf{v}}_{ts}^*]=L_{ts}T_{ts}^{-1}\), \([{\textbf{v}}_{ts}]=L_{ts}T_{ts}^{-1}\), and \([{\textbf{u}}_{ts}]=L_{ts}\), which on substitution of \(\beta ^{-1}L_{ts}= L_{ps}\) and \(g^{-1}T_{ts}=T_{ps}\) provide \(g \beta ^{-1} [{\textbf{v}}_{ts}^*]=L_{ps}T_{ps}^{-1}\), \(g \beta ^{-1} [{\textbf{v}}_{ts}] =L_{ps}T_{ps}^{-1}\), and \(\beta ^{-1}[{\textbf{u}}_{ts}]=L_{ps}\). The strategy adopted is to attempt to remove the effect of scaling in the dimensions of the trial space. Consider next utilisation of \(\alpha _{0}^{\rho } M_{ts}=M_{ps}\) to give \(\alpha ^\rho _0\beta ^3[\rho _{ts}]=M_{ps}L_{ps}^{-3}\) and similarly \(\alpha ^v_0g\beta ^2[\varvec{\sigma }_{ts}]\) on substitution of \(\alpha ^v_0=g\beta ^{-1}\alpha ^\rho _0\) gives \(\alpha ^\rho _0g^2\beta [\varvec{\sigma }_{ts}]\) but since \([\varvec{\sigma }_{ts}]=M_{ts}L_{ts}^{-1}T_{ts}^{-2}\) it follows that \(\alpha ^\rho _0g^2\beta [\varvec{\sigma }_{ts}]=M_{ps}L_{ps}^{-1}T_{ps}^{-2}\). Similarly, \(\alpha ^v_0g\beta ^3[ {\textbf{b}}_{ts}]\) on substitution of \(\alpha ^v_0=g\beta ^{-1}\alpha ^\rho _0\) returns \(\alpha ^\rho _0g^2\beta ^2[ {\textbf{b}}_{ts}]\) but \([{\textbf{b}}_{ts}]=M_{ts}L_{ts}^{-2}T_{ts}^{2}\), so \(\alpha ^\rho _0g^2\beta ^2[ {\textbf{b}}_{ts}]=M_{ps}L_{ps}^{-2}T_{ps}^{-2}\). The analysis confirms that \([{\textbf{v}}_{\beta }^*]=[{\textbf{v}}_{ps}^*]\), \([{\textbf{v}}_{\beta }]=[{\textbf{v}}_{ps}]\), \([\rho _{\beta }]=[\rho _{ps}]\), \([{\textbf{u}}_{\beta }]=[{\textbf{u}}_{ps}]\), \([\varvec{\sigma }_{\beta }]=[\varvec{\sigma }_{ps}]\), and \([{\textbf{b}}_{\beta }]=[{\textbf{b}}_{ps}]\), thus reaffirming that dimensions on \(\Omega _{\beta }\) and \(\Omega _{ps}\) are identical. The equivalence of \(\Omega _{\beta }\) and \(\Omega _{ps}\) dimensionally, conveniently provides an alternative approach to transport equations, where dimension scaling is used to identify fields on \(\Omega _\beta \) and their connection to the trial space \(\Omega _{ts}\). Moreover, the scaling space \(\Omega _\beta \) can be viewed abstractly and not necessarily connected to physics as shown in Ref. [31], where scale effects in finance were captured by the finite-similitude theory through dimensional considerations with \(\beta \) scaling capital.

Prior to examining the place of similitude rules, it is of interest to present stochastic differential equations on \(\Omega _\beta \). This transpires to be pretty straightforward since \(\Omega _\beta \) and \(\Omega _{ps}\) coincide dimensionally and equations in \(\Omega _\beta \) can therefore be approached in a traditional manner. In the case of SDEs, it is usual to start with a probability space \(({\hat{\Omega }}_\beta , \hat{{\mathcal {F}}}_\beta , \hat{\mathbb {P}}_\beta )\), where the significance of the hat symbol is purely for the purposes of distinguishing between symbols used in the text. As usual, \({\hat{\Omega }}_\beta \) is the sample space of all possible elementary outcomes, \(\hat{{\mathcal {F}}}_\beta \) is a sigma algebra (a collection of events of interest satisfying certain conditions [33]), and \(\hat{\mathbb {P}}_\beta \) is a probability measure on \(\hat{{\mathcal {F}}}_\beta \) such that \(\hat{\mathbb {P}}_\beta ({\hat{\Omega }}_\beta )=1\). Consequently, a probability space is a measure space where the measure of the whole space is equal to one. A stochastic process is defined here by the parametrized family \(\hat{\textbf{X}}_\beta =(\hat{\textbf{X}}_\beta (t_\beta ))_{t_\beta \ge 0}\) of random variables on the probability space taking values in \(\Omega _\beta \) but can also be viewed as a random function from \({\hat{\Omega }}_\beta \times [0,\infty )\rightarrow \Omega _\beta \). Equally, a transient random field involving spatial coordinate functions \({\textbf{x}}_\beta \in \Omega _\beta \) takes the form \(\hat{\textbf{X}}_\beta ({\textbf{x}}_\beta , t_\beta )\) with a corresponding stochastic-like process \(\hat{\textbf{X}}_\beta ({\textbf{x}}_\beta )=(\hat{\textbf{X}}_\beta ({\textbf{x}}_\beta , t_\beta ))_{t_\beta \ge 0}\). The focus here is on continuous stochastic processes with continuous sample paths \(t_\beta \mapsto \hat{\textbf{X}}_\beta ({\hat{\omega }}_\beta ,t_\beta )\) with \({\hat{\omega }}_\beta \in {\hat{\Omega }}_\beta \). In the application of stochastic calculi, the most useful class of stochastic processes is (continuous) semi-martingales. These are defined on a filtered probability space \(({\hat{\Omega }}_\beta , \hat{{\mathcal {F}}}_\beta , \hat{{\mathcal {F}}}_\beta ^{t_\beta }, \hat{\mathbb {P}}_\beta )\) with \(\hat{\textbf{X}}_\beta (t_\beta )\) possessing the decomposition \(\hat{\textbf{X}}_\beta (t_\beta )=\hat{\textbf{M}}_\beta (t_\beta )+\hat{\textbf{V}}_\beta (t_\beta )\), where \(\hat{\textbf{M}}_\beta (t_\beta )\) is a continuous local martingale and \(\hat{\textbf{V}}_\beta (t_\beta )\) is a stochastic process with finite variation on any finite interval [33]. Stochastic processes can be classified into practically useful types, which includes random point, stationary, translational, ergodic, Gaussian, It\(\hat{\textrm{o}}\), and diffusion processes. Diffusion-type processes are of particular interest being well researched and can be conveniently written for both Stratonovich and It\(\hat{\textrm{o}}\) calculi, and as well as a Langevin equation, i.e. where \(\hat{\textbf{B}}_\beta (t_\beta )\) is standard Brownian motion such that \(\hat{\textbf{B}}_\beta (t_\beta )-\hat{\textbf{B}}_\beta (s_\beta ) \sim {\mathcal {N}}(0,t_\beta -s_\beta )\) with \(t_\beta -s_\beta >0\), \(\hat{\textbf{a}}_\beta \) is a column matrix (of dimension \(n_\beta \)), and \(\hat{\textbf{b}}_\beta \) is a matrix (of dimension \(n_\beta \times d_\beta \)), and where \(\circ \textrm{d}\hat{\textbf{B}}_\beta \) is the Stratonovich differential, and \(\varvec{\zeta }_\beta \) is white noise, which can be viewed (somewhat imprecisely) as the time derivative of \(\hat{\textbf{B}}_\beta \) (imprecise because \(\hat{\textbf{B}}_\beta \) is nowhere differentiable).

All these equations are in a form that is essentially shorthand notation for the corresponding integral representations, where the integrals in these equations are evaluated as particular limits (see Ref. [33] for details), but in a nutshell, on a partition, \(\hat{\textbf{b}}_\beta (t_\beta )\cdot \textrm{d}\hat{\textbf{B}}_\beta (t_\beta )\) at \([t_\beta ^i,t_\beta ^{i+1}]\) is \(\hat{\textbf{b}}_\beta (t_\beta ^i) \cdot (\hat{\textbf{B}}_\beta (t_\beta ^{i+1})-\hat{\textbf{B}}_\beta (t_\beta ^i))\) and \(\hat{\textbf{b}}_\beta (t_\beta )\circ \textrm{d}\hat{\textbf{B}}_\beta (t_\beta )\) is \(\tfrac{1}{2}(\hat{\textbf{b}}_\beta (t_\beta ^i)+\hat{\textbf{b}}_\beta (t_\beta ^{i+1})) \cdot (\hat{\textbf{B}}_\beta (t_\beta ^{i+1})-\hat{\textbf{B}}_\beta (t_\beta ^i))\), and where the Langevin equation (i.e. Eq. (4c)), on integration, is selected to be one of these forms.

Numerical solutions to SDEs are generally required but consideration of the differential \(df_\beta (\hat{\textbf{X}}_\beta , t_\beta )\) (for sufficiently smooth \(f_\beta \)) provides a route to possible analytical solutions but also to deterministic components of a solution. This differential takes the forms for Stratonovich’s approach and by It\(\hat{\textrm{o}}\)’s Lemma, respectively, where the \(\tfrac{\partial f_\beta }{\partial \hat{\textbf{X}}_\beta }\) is the gradient and \(\tfrac{\partial ^2 f_\beta }{\partial \hat{\textbf{X}}_\beta ^2}\) is the Hessian, and where on contrasting the two equations provideswhich is shorthand for a corresponding integral identity, and where tr is the trace operator.

Since the two approaches can be related, focus here moves to It\(\hat{\textrm{o}}\)’s process, which is recognised as Markovian [33]. It is of interest to select \(f_\beta \) to shift focus away from the particular stochastic response to the behaviour of the underpinning probability distributions, and consequently to other statistical measures, such as means, and moments, along with deterministic aspects.

Diffusion stochastic processes have associated with second-order partial differential equations that are completely deterministic and inform on the expected value of a function (Kolmogorov backward), the probability density function (Fokker-Planck), along with a useful generalisation of the Kolmogorov backward equation, i.e. the Feynman–Kac formula [50]. These forms are returned on initial consideration of the probability measure,which is redeemable from the standard joint distribution function \(\hat{\mathbb {P}}_\beta (\hat{\textbf{x}}_\beta ^1\le \hat{\textbf{x}}_\beta ^1,...,\hat{\textbf{x}}_\beta ^n\le \hat{\textbf{x}}_\beta ^n)\) and note the corresponding probability density function (pdf) \(\hat{p}_\beta (\hat{\textbf{X}}_\beta , t_\beta )\) is defined by \(\hat{\mathbb {P}}_\beta (\hat{\textbf{X}}_\beta (t_\beta )\in \textbf{d}\hat{\textbf{x}}_\beta )=\hat{p}_\beta (\hat{\textbf{X}}_\beta , t_\beta )\textbf{d}\hat{\textbf{x}}_\beta \), and for a continuous Markovian process, the transitional (conditional) probability \(\hat{\mathbb {P}}_\beta (\hat{\textbf{X}}_\beta ,t_\beta |\hat{\textbf{y}}_\beta ,s_\beta )=\hat{\mathbb {P}}_\beta (\hat{\textbf{X}}_\beta (t_\beta )=\hat{x}_\beta |\hat{\textbf{X}}_\beta (s_\beta )=\hat{\textbf{y}}_\beta )\) with corresponding pdf \(\hat{p}_\beta (\hat{\textbf{X}}_\beta ,t_\beta |\hat{\textbf{y}}_\beta ,s_\beta )\) satisfying \(\hat{p}_\beta (\hat{\textbf{X}}_\beta ,t_\beta |\hat{\textbf{y}}_\beta ,s_\beta )\textbf{d}\hat{\textbf{x}}_\beta =\hat{\mathbb {P}}_\beta (\hat{\textbf{X}}_\beta (t_\beta )\in \textbf{d}\hat{\textbf{x}}_\beta | \hat{\textbf{X}}_\beta (s_\beta )=\hat{\textbf{y}}_\beta )\). The backward and forward equations are obtained via It\(\hat{\textrm{o}}\)’s differential Eq. (6b) on integration between the limits \(s_\beta \) and \(t_\beta \) and on consideration of the conditional expectation \(\hat{\mathbb {E}}_\beta ^{\hat{\textbf{y}}_\beta ,s_\beta }f_\beta (\hat{\textbf{X}}_\beta (t_\beta ))=\hat{\mathbb {E}}_\beta [f_\beta (\hat{\textbf{X}}_\beta (t_\beta ))|\hat{\textbf{X}}_\beta (s_\beta )=\hat{\textbf{y}}_\beta ]\), withwhere the linear operator \({\mathcal {L}}_\beta \) is where it is recognised from the probability literature that the generator \({\mathcal {A}}_\beta \) for a diffusion process is obtained fromshould this limit exist and it is readily shown that \({\mathcal {A}}_\beta ={\mathcal {L}}_\beta \) for smooth \(f_\beta \) (\(f_\beta \in L^{\infty }\) is sufficient in fact).

Setting \(\hat{u}_\beta (\hat{\textbf{y}}_\beta , s_\beta )=\hat{\mathbb {E}}_\beta ^{\hat{\textbf{y}}_\beta ,s_\beta } f_\beta (\hat{\textbf{X}}_\beta (t_\beta ))\) in place of \(f_\beta \) in Eq. (6b) leads to the backward equation \(\tfrac{\partial }{\partial s_\beta }\hat{u}_\beta +{\mathcal {L}}_\beta \hat{u}_\beta =0\), \(s_\beta <t_\beta \) and end condition \(\hat{u}_\beta (\hat{\textbf{y}}_\beta ,t_\beta )=f_\beta (\hat{\textbf{y}}_\beta )\), along with \(\tfrac{\partial }{\partial s_\beta }\hat{p}_\beta +{\mathcal {L}}_\beta \hat{p}_\beta =0\), \(s_\beta <t_\beta \), and \(\hat{p}_\beta (\hat{\textbf{X}}_\beta ,t_\beta |\hat{\textbf{y}}_\beta ,t_\beta )=\delta (\hat{\textbf{X}}_\beta -\hat{\textbf{y}}_\beta )\), where \(\delta \) is the Dirac delta function, which arises from the requirement to satisfy the end condition. Time homogeneous backward equations are similar and take the form \(\tfrac{\partial }{\partial t_\beta }\hat{u}_\beta -{\mathcal {L}}_\beta \hat{u}_\beta =0\), on setting \(\hat{u}_\beta (\hat{\textbf{X}}_\beta , t_\beta )=\hat{\mathbb {E}}_\beta ^{\hat{\textbf{X}}_\beta } f_\beta (\hat{\textbf{X}}_\beta (t_\beta ))=\hat{\mathbb {E}}_\beta [f_\beta (\hat{\textbf{X}}_\beta (t_\beta ))|\hat{\textbf{X}}_\beta (0)=\hat{\textbf{X}}_\beta ]\), with initial condition \(\hat{u}_\beta (\hat{\textbf{X}}_\beta ,0)=f(\hat{\textbf{X}}_\beta )\). The forward equation is related to the backward one through the \(L^2\)-inner product identity \(\langle {\mathcal {L}}_\beta f_\beta , g_\beta \rangle =\langle f_\beta , {\mathcal {L}}_\beta ^* g_\beta \rangle \), signifying that \({\mathcal {L}}_\beta ^*\) is the adjoint of \({\mathcal {L}}_\beta \). By means of integration by parts therefore, it is readily confirmed thatwhich on setting \({\hat{\rho }}_\beta (\hat{\textbf{X}}_\beta , t_\beta )\) to be the probability density for \(\hat{\textbf{X}}_\beta \) at \(t_\beta \) gives the forward equation \(\tfrac{\partial }{\partial t_\beta }{\hat{\rho }}_\beta -{\mathcal {L}}_\beta ^* {\hat{\rho }}_\beta =0\), with known \({\hat{\rho }}_\beta ^0(\hat{\textbf{X}}_\beta )={\hat{\rho }}_\beta (\hat{\textbf{X}}_\beta , 0)\), along with \(\tfrac{\partial }{\partial t_\beta }\hat{p}_\beta -{\mathcal {L}}_\beta ^* \hat{p}_\beta =0\), where \(\hat{p}_\beta (\hat{\textbf{X}}_\beta ,s_\beta |\hat{\textbf{y}}_\beta ,s_\beta )=\delta (\hat{\textbf{X}}_\beta -\hat{\textbf{y}}_\beta )\), with \({\hat{\rho }}_\beta \) and \(\hat{p}_\beta \) related by \({\hat{\rho }}_\beta (\hat{\textbf{X}}_\beta ,t_\beta )=\int \hat{p}_\beta (\hat{\textbf{X}}_\beta ,t_\beta |\hat{\textbf{y}}_\beta ,0){\hat{\rho }}_\beta ^0(\hat{\textbf{y}}_\beta )\textbf{d}\hat{\textbf{y}}_\beta \).

Although subscript \(\beta \) is used on all terms throughout this section, it is transparent that \(\beta -\)independence applies to Brownian processes but it is useful first to introduce the similitude rules prior to exploring the explicit dependencies in detail.

Although there exist an infinite number of similitude rules, it is the rules of low order that have practical value. The rules of interest here are zeroth-, first-, and second-order finite-similitude rules, which are of the form: which can be integrated to return of exact similitude identities.

All the rules presented in Eq. (14) integrate to give where where it is appreciated that the functions \(\alpha ^\psi _1\) and \(\alpha ^\psi _2\) are indeterminate, which means that \(R^\psi _{1,1}\), \(R^\psi _{1,2}\), and \(R^\psi _{2}\) are parameters that can be set to the advantage of the scaled model.

Similar rules to the identities above are derivable for the analysis of scaled components, which take a very similar form, where in this case the functional form of \(\alpha ^\psi _1\) and \(\alpha ^\psi _2\) is required, which permits the functions \(H^\psi _{1}\) and \(H^\psi _{2}\) to be set by means of the following proposition.

Diffusion systems suffering noise describable by standard Brownian motion \(\hat{\textbf{B}}_\beta (t_\beta )\) can be readily captured under scaling should the Brownian motion be zeroth order, i.e. independent of \(\beta \). This happens to be the case since standard Brownian motion satisfies \(\hat{\textbf{B}}_\beta (t_\beta )-\hat{\textbf{B}}_\beta (s_\beta ) \sim {\mathcal {N}}(0,t_\beta -s_\beta )\) with \(t_\beta -s_\beta >0\), and the temporal terms \(s_\beta \) and \(t_\beta \) (by arranging the dimension of time \(T_{\beta }\) to be zeroth order) are independent of \(\beta \). It follows immediately that the order of the diffusion processes captured by Eq. (4) is dictated by the order of the coefficients \(\hat{\textbf{a}}_\beta \) and \(\hat{\textbf{b}}_\beta \). Moreover, Eq. (4) (or Eq. (5)) reveals that the order of \(\textrm{d}\hat{\textbf{X}}_\beta \) is the maximum of the orders of \(\hat{\textbf{a}}_\beta \) and \(\hat{\textbf{b}}_\beta \) since the products \(\hat{\textbf{b}}_\beta \circ \textrm{d}\hat{\textbf{B}}_\beta \) and \(\hat{\textbf{b}}_\beta \cdot \textrm{d}\hat{\textbf{B}}_\beta \) have the same order as \(\hat{\textbf{b}}_\beta \) with zeroth-order \(\hat{\textbf{B}}_\beta \). Note that the order of forward and backward equations (even if undefined) discussed in Sec. 3.3 is of no concern since these ultimately are dependent on \(\hat{\textbf{X}}_\beta \) (and \(t_\beta \)), so they are well defined using similitude rules on construction of \(\hat{\textbf{X}}_{ps}\) in the physical space.

Bild vergrößern

To demonstrate how scaling analysis and scaled experimentation can be applied to a noisy physical mechanical system, the damped-vibratory system depicted in Fig. 2 is investigated.

One of the attractive benefits of using the scaling space \(\Omega _\beta \) is that it possesses all the hallmarks of a physical space and consequently governing equations of any mathematical type can be immediately formulated in the space. The system depicted in Fig. 2 is governed by the differential equation,where \(\mu _\beta \) is the friction coefficient, \(\theta _\beta \) is slope angle, \(m_\beta \) is mass, \(c_\beta \) is the damping coefficient, \(k_\beta \) is stiffness, and \(G_\beta \) is acceleration due to gravity.

A feature of the system is a noisy forcing term \(F_\beta \) making Eq. (21) a Langevin-type equation, and consequently, consideration must be given to the how the noise is to be interpreted. An It\(\hat{\textrm{o}}\) approach is adopted with a forcing differential relationship of the form,where \(F_\beta ^0\) is a temporally invariant force amplitude and \(\epsilon _\beta \) is a scaling parameter associated to the standard Brownian motion \(\hat{B}_\beta (t_\beta )\).

The kinematic terms are related to corresponding terms in the trial space, with displacement \(u_\beta =\beta ^{-1}u_{ts}\), velocity \(v_\beta ={\dot{u}}_\beta =g\beta ^{-1}{\dot{u}}_{ts}\), and acceleration \(a_\beta ={\dot{v}}_\beta =\ddot{u}_\beta =g^2\beta ^{-1}\ddot{u}_{ts}\). Note then it is possible on setting \(\hat{X}_\beta ^1=u_\beta \) and \(\dot{\hat{X}}_\beta ^1=\hat{X}_\beta ^2=v_\beta \) (other choices are possible) to reformulate Eq. (21) into the standard diffusion form, which on contrasting with Eq. (4) means that Prior to proceeding with the solution of this system, it is necessary to enforce physical constraints imposed by the real system as presented in the trial space. Replica scaling is assumed with material types unchanged in scaled models, although allowance is made for switching the hydraulic fluid in the damper as required. Invariant friction \(\mu _{\beta }=\mu _{ts}=\mu _{ps}\) and slope \(\theta _\beta =\theta _{ts}=\theta _{ps}\) conditions are enforced. The projection from the trial space \(\Omega _{ts}\) yields the relationships \(m_\beta =\alpha _{0}^\rho m_{ts}\), \(k_\beta =\alpha _{0}^\rho g^2 k_{ts}\), \(c_\beta =\alpha _{0}^\rho g c_{ts}\), \(G_\beta =g^2\beta ^{-1}G_{ts}\), \(\epsilon _{\beta }=g^{-1}\epsilon _{ts}\), and \(F_\beta ^0=\alpha _{0}^\rho g^2 \beta ^{-1} F_{ts}^0\). The invariance of density \(\rho _{\beta }=\alpha _{0}^\rho \beta ^3 \rho _{ts} = \rho _{ps}\) necessitates \(\alpha _{0}^\rho =\beta ^{-3}\), and consequently, \(m_\beta =\beta ^{-3} m_{ts}\), \(k_\beta =\beta ^{-3} g^2 k_{ts}\), \(c_\beta =\beta ^{-3} g c_{ts}\), and \(F_\beta ^0=g^2 \beta ^{-4} F_{ts}^0\) with the relationship for acceleration due to gravity unaffected. It is convenient at this point to set \(g=\beta \) and this ensures \(k_\beta =\beta ^{-1}k_{ts}\), and given that replica scaling of springs satisfies \(k_{ps}=\beta ^{-1}k_{ts}\) [43], this setting conveniently provides the zeroth-order condition \(k_\beta =k_{ps}\). Consequently, \(c_\beta =\beta ^{-2} c_{ts}\), \(F_\beta ^0= \beta ^{-2} F_{ts}^0\), and \(G_\beta =\beta G_{ts}\) with the damper requiring a change in the hydraulic fluid (see reference [43] for details) to give \(c_{ps}=\beta ^{-2} c_{ts}\) providing the zeroth-order relationship \(c_\beta = c_{ps}\). It is assumed further that the condition \(F_{ps}^0= \beta ^{-2} F_{ts}^0\) can be set by arrangement so that \(F_\beta ^0= F_{ps}^0\) but unfortunately the relationship for acceleration due to gravity \(G_\beta =\beta G_{ts}\) does not (for \(\beta \not = 1\)) equal \(G_{ps} = 9.81 \mathrm {m/s^2}\). Consequently, despite changing the hydraulic fluid in the damper, the system depicted in Fig. 2 as described by Eq. (21) cannot be captured by a single scaled experiment.

Consideration is now given to the possibility that a first-order finite-similitude identity is applicable with attention on acceleration due to gravity \(G_\beta \). Consider then the first-order identity \(\tfrac{d}{d\beta }(\alpha _1\tfrac{d}{d\beta }{\textbf{a}}_\beta )\equiv {\textbf{0}}\) or equivalently \(\tfrac{d}{d\beta }(\alpha _1\tfrac{d}{d\beta }\beta ){\textbf{a}}_{ts}\equiv {\textbf{0}}\) with \({\textbf{a}}_{ts}\) assumed invariant. This relationship is satisfied with \(\alpha _1=1\) (recall the \(\alpha _1(1)=1\)) and note for Prop. 4.1 that \(\tfrac{dH_1}{d\beta } \alpha _1=-1\), which solves to give \(H_1=1-\beta \) with \(H_1(1)=0\) as required. Consequently, the first-order identity \({\textbf{a}}_{1}={\textbf{a}}_{\beta }+H_1\alpha _1{\textbf{a}}_\beta ^{\prime }\) reduces to \({\textbf{a}}_{1}={\textbf{a}}_{\beta }+(1-\beta )\alpha _1{\textbf{a}}_\beta ^{\prime }\) with \(\alpha _1=1\). Similar identities apply to velocity and displacement. An additional diffusion process is required in scaling analysis which flows from differentiation of Eq. (21) and Eq. (22) with respect to \(\beta \) to give with corresponding diffusion equations, where the ratio \(\tfrac{\hat{X}_\beta ^2}{|\hat{X}_\beta ^2|}\) is taken to be independent of \(\beta \) for this analysis.

It is simple matter to confirm that systems Eq. (23) and Eq. (26) can be combined to capture the response of the full-size system with noise included. To confirm this, note that which on substitution of the first-order identities \(\hat{X}_1^1=\hat{X}_\beta ^1+H_1\alpha _1(\hat{X}_\beta ^1)^{\prime }\), \(\hat{X}_1^2=\hat{X}_\beta ^2+H_1\alpha _1(\hat{X}_\beta ^2)^{\prime }\), \(G_1=G_\beta +H_1\alpha _1G_\beta ^{\prime }\), \(\epsilon _1=\epsilon _\beta +H_1\alpha _1\epsilon _\beta ^{\prime }\) with \(H_1=1-\beta \) and \(\tfrac{\hat{X}_\beta ^2}{|\hat{X}_\beta ^2|}=\tfrac{\hat{X}_1^2}{|\hat{X}_1^2|}\) returns the full-size system of equations.

Note, however, the identity \(\epsilon _1=\epsilon _\beta +H_1\alpha _1\epsilon _\beta ^{\prime }\) is at best approximate, although satisfied certainly under the zeroth-order assumption for \(\epsilon _\beta \), i.e. with \(\epsilon _{ts}\) satisfying \(\epsilon _{ps}=\beta ^{-1}\epsilon _{ts}\). Additionally, however, the identity \(\epsilon _{ps}=\beta ^{-2}\epsilon _{ts}\) (if applicable) also satisfies the first-order requirements as readily confirmed on substitution into \(\epsilon _1=\epsilon _\beta +H_1\alpha _1\epsilon _\beta ^{\prime }\). Things are relatively more straightforward in scaled experimentation with the first-order expression for acceleration due to gravity \({G}_1={G}_{\beta _1}+R_1({G}_{\beta _1}-{G}_{\beta _2})\) with \(G_\beta =\beta G_{ts}\) providing \(R_1=(1-\beta _1)/(\beta _1-\beta _2)\) to secure the necessary invariance in gravity. Consequently, the identities \(\hat{X}_1^1=\hat{X}_{\beta _1}^1+R_1(\hat{X}_{\beta _1}^1-\hat{X}_{\beta _2}^1)\), \(\hat{X}_1^2=\hat{X}_{\beta _1}^2+R_1(\hat{X}_{\beta _1}^2-\hat{X}_{\beta _2}^2)\), \(G_1=G_{\beta _1}+R_1(G_{\beta _1}-G_{\beta _2})\), and with \(\epsilon _{ps}=\epsilon _1=\epsilon _{\beta _1}+R_1(\epsilon _{\beta _1}-\epsilon _{\beta _2})\) assumed, return the response of the full-size system as seen on substitution into Eq. (23).

In order to assess the suitability of first-order similitude rule in a noisy environment, the system depicted in Fig. 2 is analysed by means of the Monte Carlo method. Of interest here are the expected response \(\hat{\mathbb {E}}_\beta [\hat{{\textbf{X}}}_\beta ]\) along with the variance \(\hat{\mathbb {V}}_\beta [\hat{{\textbf{X}}}_\beta ]\) as predicted from Eq. (23) and contrasted with returns from the Monte Carlo simulation. Observe that \(d\hat{\mathbb {E}}_\beta [\hat{{\textbf{X}}}_\beta ]=\hat{{\textbf{a}}}_\beta dt_\beta \) (obtained on setting \(f_\beta =\hat{\mathbb {E}}_\beta [\hat{{\textbf{X}}}_\beta ]\) in Eq. (6b)) which signifies that the original noise-free system of differential equations provides the expected value of \(\hat{{\textbf{X}}}_\beta \). The variance returned from the SDEs is provided on consideration of which are differential equations that can be readily solved on solving first \(d\hat{\mathbb {E}}_\beta [\hat{{\textbf{X}}}_\beta ]=\hat{{\textbf{a}}}_\beta dt_\beta \), but note the absence of spread in \(\hat{X}_\beta ^1\) arising because \(b_\beta ^1\) is set to zero.

The models analysed include the full-scale model (\(\beta _0=1\)), trial model 1 (\(\beta _1=2/3\)), and trial model 2 (\(\beta _2=1/3\)), details of which are presented in Table 1. The size of the forcing term amplitude is chosen to be sufficiently large to have an impact but not an overriding impact on the deterministic element of the diffusion system.

The Monte Carlo simulation involved 500 realisations for each of the spaces (i.e. full-scale, trial spaces 1 and 2) providing the raw data for algebraic determination of means and variances. The governing system of differential satisfies the first-order rule (along with the SDEs), which translates into the mean \(\hat{\mathbb {E}}_\beta [\hat{{\textbf{X}}}_\beta ]\) and standard deviation \(\hat{\mathbb {S}}_\beta [\hat{{\textbf{X}}}_\beta ]=\sqrt{\hat{\mathbb {V}}_\beta [\hat{{\textbf{X}}}_\beta ]}\) returned also satisfying this rule. It is expected therefore that the small models will replicate behaviours at full size as represented by the SDEs. It is of interest to discover however, how the data randomly generated through the Monte Carlo method respond in this regard, which provides the returns presented in Table 2. The average difference between the computations at full scale and combinations of trial space projections show a percentage error in displacement smaller than 2.72%, so providing a good match. The mean absolute error (MAE) is \(3.07\,\textrm{mm}\) or 26.9% for zeroth-order \(\epsilon _\beta \) and \(2.037\,\textrm{mm}\) or 17.69% if first-order \(\epsilon _\beta \) is assumed. The standard deviation of the error is \(4.63\,\textrm{mm}\) and \(3.23\,\textrm{mm}\) for zeroth-order and first-order \(\epsilon _\beta \), respectively. Effectively, 95% of errors between the trial approach and full scale will not be greater than \(9.1\,\textrm{mm}\). Although this result demonstrates a good approximation of the full-scale behaviour by both zeroth and first-order rules, statistical analysis of the means and spread is necessary to ensure that differences are sufficiently small at each time step. The difference between the MAE of the means at each time step is less than \(1.5\,\textrm{mm}\) (10.88% percent error) and \(0.2\,\textrm{mm}\) (or 16.44%) for the zeroth- and first-order cases, respectively, and similarly, the percentage error in average deviation values is 7.34% and 14.13%. Finally, the comparison of analytically derived expressions for the variance of the velocity \(\hat{X^2}\) and that provided by Monte Carlo simulation presents a maximum difference of \(0.5\,\mathrm {mm/s}\) across all spaces, confirming consistency in the two approaches. For the percentage error calculations, values smaller than 0.5% of the measurement range \([-63,100]\,\textrm{mm}\) were not considered.

The visualisation of the stochastic simulations is presented in Fig. 3 illustrating the efficacy of combining the projected trial space predictions. Fig. 3(a) and Fig. 3(b), respectively, show the error bands using the standard deviation as a measure, 68% (1 standard deviation) of projections will be within the red and orange dashed bands and 95% (2 standard deviations) within the green and yellow dashed bands. Similarly, Fig. 3c and d presents the visualisations of projected trial space combination (virtual model) and full-scale space realisations (copper colour with a transparency parameter of 0.2) compared to a random realisation (blue), for the zeroth-order \(\epsilon _\beta \) case. The regions with deep colouring represent the highest concentration of values from the simulated values demonstrating a close match between the virtual (combination of trial space projections) and the full-scale model.

It is of interest to examine the scaling of a system that features noise in both the mechanical and electrical subsystems. The starting point for any scaling analysis is description of the system under consideration in the scaling space \(\Omega _\beta \). The system investigated (previously analysed in Ref. [51]) is presented in Fig. 4, consisting of a moving mass \(m_\beta \) attached to spring of stiffness \(k_\beta \) with movement damped by a magnetic-coil damper characterised by a force-coupling constant \({\mathcal {B}}_\beta \). The spring and damper are attached to a moving based that is externally excited with displacement \(x_\beta ^0\) and velocity \({\dot{x}}_\beta ^0\). The excitation of the base is noisy, captured here through forcing-like terms of the form, where \(X_\beta ^0\) is the displacement amplitude, \(\epsilon _\beta \) is the scaling parameter for \(\hat{B}_\beta \) the standard Brownian motion, and where an It\(\hat{\textrm{o}}\) interpretation is applied.

The external force \(F_\beta \) is incorporated here to ensure that motion of the base is consistently described by either \({x}_\beta ^0\) or its derivative \({\dot{x}}_\beta ^0\) in the presence of white noise. The coupling between electrical and mechanical functions is through the action of the magnetic damper, which is presented in Fig. 4, consisting of a permanent axial magnet, ferromagnetic poles, coil, shell, non-magnetic supports, and a movable rod. The coil and permanent magnet couples mechanical and electrical forces with the constant \({\mathcal {B}}_\beta \) representing the strength of the coupling between them. This coupling constant arises out of the active length of the coil exposed to the magnetic field. The two contours presented in the figure show the path taken by the magnetic field, which purposely crosses the coils, perpendicularly. The system in its simplest form can be captured mathematically by a model with the two degrees of freedom of electrical charge \(Q_\beta ^f\) and mass displacement \(x_\beta \). The physical realisation of the scaling space \(\Omega _\beta \) means that governing equations describing the dynamics of the coupled system can be immediately formulated in terms of mass \(m_\beta \), stiffness \(k_\beta \), inductance \(L_\beta \), capacitance \(C_\beta \), and resistance \(R_\beta \). To arrive at the correct representations, it is first convenient to transform Maxwell equations into integral form on \(\Omega _\beta \), i.e. formulate where \(I^f_\beta =\int _{ S_\beta }{\textbf{J}}^f_\beta \cdot {\textbf{n}}_\beta dS_\beta \) is the current passing through surface \(S_\beta \) with boundary \({\mathcal {C}}_\beta \), \(Q^f_\beta =\int _{{\mathcal {V}}_\beta } \rho ^f_\beta dV_\beta \) is the charge in domain \({\mathcal {V}}_\beta \), and electromotive force (emf) defined to be \(\mathcal {E}_\beta =\int _{{\mathcal {C}}_\beta } {\textbf{E}}_\beta \cdot \mathbf {d \ell }_\beta \) with magnetic flux by \(\Phi _\beta = \int _{ S_\beta } {\textbf{B}}_\beta \cdot {\textbf{n}}_\beta dS_\beta \).

Eq. (32c) (Faraday’s law) along with Lorentz force density \(f^{f}_\beta ={\textbf{J}}^{f}_\beta \times {\textbf{B}}_\beta \) is important here. Consider then one active loop of the coil with the magnetic field \({\textbf{B}}_\beta \) lying in its plane. Accounting for the relative movement of the permanent magnet to the coil in \(\mathcal {E}_\beta =\int _{{\mathcal {C}}_\beta } {\textbf{E}}_\beta \cdot \mathbf {d \ell }_\beta \) is achieved on swapping \({\textbf{E}}_\beta \) with \({\textbf{E}}_{\beta }^*\), where \({\textbf{E}}_{\beta }^*={\textbf{E}}_{\beta }+{\textbf{v}}_{\beta }\times {\textbf{B}}_{\beta }\) to return \(\mathcal {E}_\beta =\int _{{\mathcal {C}}_\beta } ({\textbf{E}}_{\beta }+{\textbf{v}}_{\beta }\times {\textbf{B}}_{\beta }) \cdot \mathbf {d \ell }_\beta \). Consequently, for one loop of the coil, this relationship provides the emf \(2\pi {\bar{R}}_\beta \textrm{B}_\beta ({\dot{x}}_\beta -{\dot{x}}_\beta ^0)\), so for \(N_\beta ^{active}\) active coil loops (i.e. those cutting the magnetic field), the emf voltage across the circuit is \(V_\beta =2\pi {\bar{R}}_\beta N_\beta ^{active} \textrm{B}_\beta ({\dot{x}}_\beta -{\dot{x}}_\beta ^0)={\mathcal {B}}_\beta ({\dot{x}}_\beta -{\dot{x}}_\beta ^0)\), where \({\dot{x}}_\beta \) is the speed of the mass \(m_\beta \) in Fig. 4, \({\dot{x}}_\beta ^0\) is the base velocity, \({\bar{R}}_\beta \) is the mean radius of the coil, and \({\mathcal {B}}_\beta =2\pi {\bar{R}}_\beta N_\beta ^{active} \textrm{B}_\beta \). Similarly, the elemental damping force provided by the Lorentz relationship is given by \(dF_\beta ^{damp}=I_\beta ^f \mathbf {d \ell }_\beta \times {\textbf{B}}_\beta \), which means for \(N_\beta ^{active}\) active loops that the damping force is \(F_\beta ^{damp}={\mathcal {B}}_\beta I_\beta ^f\), where \({\mathcal {B}}_\beta =2\pi {\bar{R}}_\beta N_\beta ^{active} \textrm{B}_\beta \). It follows that the governing electromechanical equations for the system depicted in Fig. 4 with circuits outlined in Fig. 5 are providing a coupled system of Langevin-type differential equations, with \(I_\beta ^f={\dot{Q}}_\beta ^f\) where \({\dot{Q}}_\beta ^f\) signifying the rate of charge, and the forcing terms on the left hand side of these equations represented in Eq. (31).

Prior to investigating the solution and scaling of these equations, it is necessary to relay physical constraints projected from the trial space. Mechanical physical quantities satisfy \(m_\beta =\alpha _0^\rho m_{ts}\), \(F_\beta =\alpha _0^v g F_{ts}=\alpha _0^\rho g^2 \beta ^{-1} F_{ts}\) (since \(\alpha _0^v=g\beta ^{-1}\alpha _0^\rho \)), and \(k_\beta =\alpha _0^\rho g^2 k_{ts}\), with kinematic relationships, \(x_\beta =\beta ^{-1}x_{ts}\), \({\dot{x}}_\beta =g\beta ^{-1}{\dot{x}}_{ts}\), and \(\ddot{x}_\beta =g^2\beta ^{-1}\ddot{x}_{ts}\). Similarly, electromagnetic charge quantities satisfy \(Q_\beta ^f=\alpha ^G_0 g Q_{ts}^f\), \({\dot{Q}}_\beta ^f=\alpha ^G_0 g^2 {\dot{Q}}_{ts}^f\) (equivalently \(I_\beta ^f=\alpha ^G_0 g^2 I_{ts}^f\)), and \({\ddot{Q}}_\beta ^f=\alpha ^G_0 g^3 {\ddot{Q}}_{ts}^f\), with \(\alpha _0^G L_\beta =\alpha _0^F g^{-2} \beta L_{ts}\), \(\alpha _0^G R_\beta =\alpha _0^F g^{-1}\beta R_{ts}\), \(\alpha _0^G C_\beta ^{-1}=\alpha _0^F \beta C_{ts}^{-1}\) (or \(\alpha _0^F C_\beta =\alpha _0^G \beta ^{-1} C_{ts}\)). The relationship for \({\mathcal {B}}_\beta \) can be obtained using several approaches with one being the direct substitution of \({\bar{R}}_\beta =\beta ^{-1}{\bar{R}}_{ts}\), \( N_\beta ^{active}= N_{ts}^{active}\), and \(\textrm{B}_{\beta }=\alpha ^F_0 \beta ^3 \textrm{B}_{ts}\) into the relationship \({\mathcal {B}}_\beta =2\pi {\bar{R}}_\beta N_\beta ^{active} \textrm{B}_\beta \) to give \({\mathcal {B}}_\beta =\alpha _0^F\beta ^2{\mathcal {B}}_{ts}\). Another possibility is to deduce from Eq. (33b) directly that \({\mathcal {B}}_\beta =\alpha _0^F\beta ^2{\mathcal {B}}_{ts}\), and yet another is to note that Eq. (33a) provides \(\alpha _0^G{\mathcal {B}}_\beta =\alpha _0^\rho \beta ^{-1}{\mathcal {B}}_{ts}\). However, since \(\alpha ^{v}_0=\alpha ^{F}_0\alpha ^G_0 g\beta ^2\) or equivalently \(\alpha ^{\rho }_0=\alpha ^{F}_0\alpha ^G_0 \beta ^3\), which on substitution returns once again \({\mathcal {B}}_\beta =\alpha _0^F\beta ^2{\mathcal {B}}_{ts}\).

The benefit of scaling space \(\Omega _\beta \) with the direct representation of Eq. (33) is that through the application of differential finite-similitude identities, it can be deduced if the system presented in Fig. 4 scales. Should the quantities \(F_\beta , k_\beta , x_\beta ^0, m_\beta , {\mathcal {B}}_\beta \), and \(k_\beta \) for example be independent of \(\beta \) along with \(L_\beta , R_\beta \), and \(C_\beta \), then it can be readily checked that the system Eq. (33) is zeroth order. Consequently, a single scaled experiment would be sufficient although in reality, such a simple outcome is unlikely, so it is important therefore to investigate how quantities scale in practice. Under replica scaling (i.e. no change in material type), resistance, inductance, capacitance, stiffness, and mass satisfy, respectively, the following relationships: \(R_{\beta }=\rho ^{elec}_{\beta }\ell _\beta /A_{\beta }\), \(L_\beta =\mu _\beta n_\beta ^2 A_\beta \ell _\beta \), \(C_\beta =\epsilon _\beta A_\beta / \ell _\beta \), \(k_\beta =E_\beta A_\beta \ell _\beta \), and \(m_\beta =\rho ^{mech}_\beta A_\beta \ell _\beta \), with material properties \(\rho ^{elec}_{\beta }, \mu _\beta , \epsilon _\beta , E_\beta \), and \(\rho ^{mech}_\beta \) assumed here to be independent of \(\beta \). In view of the fact that area \(A_\beta =\beta ^{-2}A_{ts}\), length \(\ell _\beta =\beta ^{-1}\ell _{ts}\), and number of turns per unit length \(n_\beta =\beta n_{ts}\), it follows under replica scaling, \(R_\beta =\beta R_{ts}\), \(L_\beta =\beta ^{-1}L_{ts}\), \(C_\beta =\beta ^{-1}C_{ts}\), \(k_\beta =\beta ^{-1}k_{ts}\), and \(m_\beta =\beta ^{-3}m_{ts}\).

Although the projected mechanical relationships \(m_\beta =\alpha _0^\rho m_{ts}\) and \(k_\beta =\alpha _0^\rho g^2 k_{ts}\) provide the desired behaviours \(m_\beta =\beta ^{-3}m_{ts}\) and \(k_\beta =\beta ^{-1}k_{ts}\) on setting \(\alpha _0^\rho =\beta ^{-3}\) and \(g=\beta \), the same is not true for the electrical relationships. In this case, projected electrical relationships \(\alpha _0^G L_\beta =\alpha _0^F g^{-2} \beta L_{ts}\), \(\alpha _0^F C_\beta =\alpha _0^G \beta ^{-1} C_{ts}\), and \(\alpha _0^G R_\beta =\alpha _0^F g^{-1} \beta R_{ts}\) cannot be matched simultaneously with \(L_\beta =\beta ^{-1}L_{ts}\), \(C_\beta =\beta ^{-1}C_{ts}\), and \(R_\beta =\beta R_{ts}\). This transpires because \(\alpha _0^G\) and \(\alpha _0^F\) are not independent since they satisfy \(\alpha ^{\rho }_0=\alpha ^{F}_0\alpha ^G_0 \beta ^3\) which gives \(\beta ^{-6}=\alpha ^{F}_0\alpha ^G_0\). It follows that, \((\alpha _0^G)^2 L_\beta =\alpha _0^F \alpha _0^G \beta ^{-1} L_{ts} =\beta ^{-6} \beta ^{-1} L_{ts}\) and \((\alpha _0^F)^2 C_\beta =\alpha _0^G \alpha _0^F \beta ^{-1} C_{ts} =\beta ^{-6} \beta ^{-1} C_{ts}\), giving \(\alpha _0^F=\alpha _0^G=\beta ^{-3}\). But this means that the identity \(\alpha _0^G R_\beta =\alpha _0^F g^{-1} \beta R_{ts}\) reduces to \(R_\beta =R_{ts}\) and not the replica behaviour \(R_\beta =\beta R_{ts}\). In addition, under replica scaling with the same ferromagnetic material of identical magnetic susceptibility in the scaled damper, then \({B}_\beta ={B}_{ts}\), so with \({B}_\beta =\alpha _0^F\beta ^3{B}_{ts}\) this requirement is also captured by \(\alpha _0^F=\beta ^{-3}\). It is necessary therefore to employ physical modelling for zeroth-order conditions, with the resistance of the trial space system adjusted so that the identity \(R_\beta =R_{ps}\) applies. This is achieved with \(R_{ts}=\beta ^{-1}R_{ps}\) requiring additional resistance in trial space system, which is relatively easy to arrange, experimentally. Note additionally that the coupling relationship \({\mathcal {B}}_\beta =\alpha _0^F\beta ^2{\mathcal {B}}_{ts}\) returns the zeroth-order condition \({\mathcal {B}}_\beta =\beta ^{-1}{\mathcal {B}}_{ts}\), since \({\mathcal {B}}_\beta =2\pi {\bar{R}}_\beta N_\beta ^{active} \textrm{B}_\beta =\beta ^{-1}2\pi {\bar{R}}_{ts} N_{ts}^{active} \textrm{B}_{ts}=\beta ^{-1}{\mathcal {B}}_{ts}\).

It is of interest to first capture the response of the system with noise absent under these settings but where the base displacement \(x_\beta ^0\) is dependent on \(\beta \) constrained by a first-order relationship. Consider then the imposition of a first-order length invariance on the system with the identity \(\tfrac{d}{d\beta }(\alpha _1^v\tfrac{d}{d\beta }x_\beta )\equiv 0\) with \(x_\beta =\beta ^{-1}x_{ts}\) with \(x_{ts}\) assumed momentarily to be invariant of \(\beta \). This provides \(\tfrac{d}{d\beta }(\alpha _1^v\tfrac{d}{d\beta }x_\beta )=\tfrac{d}{d\beta }(\alpha _1^v\tfrac{d}{d\beta }\beta ^{-1})x_{ts}\equiv 0\), which gives \(\alpha ^v_1=\beta ^2\) with \(\alpha ^v_1(1)=1\). It follows that two scaled experiments with base displacement satisfying,with \(R_1=(1-\beta _1^{-1})/(\beta _1^{-1}-\beta _2^{-1})\) should replicate the response of the full-scale system.

In addition, it is required that \(\alpha _1=\alpha _1^v=\alpha _1^G\), which ensures that the operator \(\tfrac{d}{d\beta }(\alpha _1\tfrac{d}{d\beta }\cdot )\) on application to the system Eq. (33) gives identically zero. The theory above is confirmed for a specific case with properties provided in Table 3 with the scaled models set at scales \(\beta _1=0.5\) and \(\beta _2=0.25\). This setting provides \({\mathcal {B}}_{ps}=\beta ^{-1}_1{\mathcal {B}}_{ts 1}=\beta ^{-1}_2{\mathcal {B}}_{ts 2}\), \(m_{ps}=\beta ^{-3}_1 m_{ts 1}=\beta ^{-3}_2 m_{ts 2}\), \(k_{ps}=\beta ^{-1}_1 k_{ts 1}=\beta ^{-1}_2 k_{ts 2}\), and \(L_{ps}=\beta ^{-1}_1 L_{ts 1}=\beta ^{-1}_2 L_{ts 2}\). Eq. (33), and Eq. (31) can be transformed into a standard diffusion form on setting \(\hat{X}_\beta ^1=x_\beta \), \(\hat{X}_\beta ^2={\dot{x}}_\beta \), \(\hat{X}_\beta ^3=Q_\beta ^f\), and \(\hat{X}_\beta ^4={\dot{Q}}_\beta ^f\) (other choices are possible) to give which means that on contrasting with Eq. (4) that where as a consequence of \(X_\beta ^0\) being first order, with all other parameters being zeroth order, provides a first-order noisy system.

Note that, since \(\epsilon _\beta =g^{-1}\epsilon _{ts}=\beta ^{-1}\epsilon _{ts}\) and with \(X_\beta ^0=\beta ^{-1}X_{ts}^0\), it can be deduced that zeroth- and first-order behaviours for \(\epsilon _\beta \) are captured with \(\epsilon _{ps}=\beta ^{-1}\epsilon _{ts}\) and \(\epsilon _{ps}=\epsilon _{ts}\), respectively. However, the product term \(X_{\beta }^0\epsilon _\beta \) in Eq. (35) constrains either \(X_{\beta }^0\) or \(\epsilon _\beta \) to be zeroth order (with the other first order) for a first-order system.

Analogous to the mechanical diffusion system analysis, the suitability of first-order similitude in noisy electromechanical environment is assessed. Again the Monte Carlo method is applied with results returned contrasted against those provided by the SDEs presented in the previous section. Similarly, the expected response \(\hat{\mathbb {E}}_\beta [\hat{{\textbf{X}}}_\beta ]\) and variance \(\hat{\mathbb {V}}_\beta [\hat{{\textbf{X}}}_\beta ]\) are of interest as predicted from Eq. (35) in contrast with returns from the Monte Carlo simulation. The mean response is once again returned form the noise-free equations, i.e. \(d\hat{\mathbb {E}}_\beta [\hat{{\textbf{X}}}_\beta ]=\hat{{\textbf{a}}}_\beta dt_\beta \), with variance obtained from \(d \hat{\mathbb {V}}_\beta [\hat{X}^i_\beta ]=0\), (\(i=1\) and \(i=3\)) and \(d \hat{\mathbb {V}}_\beta [\hat{X}^2_\beta ]=(b_\beta ^i)^2 dt_\beta \), (\(i=2\) and \(i=4\)), where again the SDEs (as formed) predict no spread in particular components of \(\hat{{\textbf{X}}}_\beta \).

The models analysed in this investigation are the full-scale model (\(\beta _0=1\)), trial model 1 (\(\beta _1=1/2\)), and trial model 2 (\(\beta _2=1/4\)), with scaled quantities and parameters provided in Table 3. The Monte Carlo simulation involved 500 realisations providing the raw data for the investigation, as presented in Fig. 6 for each space. The first-order expression derived in Eq. (34) for the displacement was used to project the trial space realisations to full size. The statistical analysis of the error between realisations of the combined trial space projections (virtual model) and realisations of the full-scale space is presented in Table 4. The analytical comparison between the full-scale and virtual values results in an error smaller than \(0.00056\,\textrm{mm}\) or 2%, where the displacement range is \([-0.1,0.1]\,\textrm{mm}\). The error between spaces in the Monte Carlo simulations returns a mean absolute error smaller than \(0.006\,\textrm{mm}\) or 30.4% and standard deviation of the error values of \(0.0073\,\textrm{mm}\), i.e. 95% of the displacement values of the virtual approximation are closer than \(0.0146\,\textrm{mm}\) from the full-scale values. Consideration of the error in \(\hat{X}^1_{SDE}-\hat{X}^1_{MC}\) for each time step returns a mean absolute error of the full scale and virtual means of less than \(0.00014\,\textrm{mm}\) (or 27.5%), with errors in \(\hat{X}^3_{SDE}-\hat{X}^3_{MC}\) smaller than \(0.00004\,\textrm{C}\) (or 37.4%). The difference between average deviation between full scale and virtual values of \(\hat{X}^1_{MC}\) is \(0.0003\,\textrm{mm}\) or 6%, and \(0.0004\,\textrm{C}\) (or 43%) for \(\hat{X}^3_{MC}\). Therefore, a good approximation of the full-scale behaviour is demonstrated.

An overall depiction of the response of the noisy electromagnetic system (depicted in Fig. 5) is presented in Fig. 6. Figure 6a and b provides a temporal-response and phase-space depiction, respectively, for means and spread at full size as returned by full-scale and virtual models. Dashed lines capture the spread with solid lines reserved for mean values. Analogously to the spring-mass-damper system, Fig. 6c and d shows returns of 500 realisations (with the transparency parameter set at 0.2) from the full-scale and virtual models (combination of trial space projections), respectively. The closeness of the results confirms the feasibility of combining trial space information in a noisy environment to predict sensible responses at full size.