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02-03-2023

Analytical Thermodynamics

Authors: Paolo Podio-Guidugli, Epifanio G. Virga

Published in: Journal of Elasticity | Issue 4-5/2023

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Abstract

This paper proposes a theory that bridges classical analytical mechanics and nonequilibrium thermodynamics. Its intent is to derive the evolution equations of a system from a stationarity principle for a suitably augmented Lagrangian action. This aim is attained for homogeneous systems, described by a finite number of state variables depending on time only. In particular, it is shown that away from equilibrium free energy and entropy are independent constitutive functions.

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At first sight, this start might not seem related to the matter at hand, but the reader will soon discover that it really is.

The same counterexample shows that in a degenerate case the path of light may neither be the shortest nor the longest.

The reader could also profit from consulting the review [37] on this subject.

Equivalently, as in Biot’s original works, ℛ could be minimized under the constraint that both generalized forces and total working are kept fixed (see [61, p. 119]). In such a formulation, this principle is also known as the principle of minimum constrained dissipation.

Or towards equilibrium, in the absence of external agencies.

Going backwards in time, these papers were preceded, essentially in the same line of thought, by [29], while the origin of the method can be retraced in an early paper by Kaufman and Morrison [39], framed in the context of the quasilinear equations of plasma physics. This paper was later followed by [38, 48, 49]. More recent applications of the method can be found in the specific fields of multiphase fluids [18] and dissipative magnetohydrodynamics [11].

A similar objective was recently pursued in [9], but without accounting for the second law of thermodynamics.

Despite the novelty of this approach, studies on the Lagrange-Hamilton formalism for nonequilibrium thermodynamics had appeared before in the literature: we mention, for example, [22] and refer the reader to the discussion in [73].

As remarked in [1, p. 44], the equations of motion generated by this principle differ from those (also called vakonomic) that are generated by requiring stationarity of the action under the nonholonomic constraint meant to restrict accessibility in phase space. Vakonomic dynamics was developed by V. V. Kozlov in a series of six papers published in the 1980’s (referenced as [329] in [1, p. 490]).

Calling this case dynamical is typical of the Italian School of Mechanics (see, for example, [43, p. 247]). Along similar lines, in the British tradition, (see, for example, [77, p. 57]), the word natural is used when the Lagrangian contains only terms of degree 2 or 0 in the velocities.

Kinetic momenta, as they are called in the Italian tradition (see, for example, [43, p. 246]), were introduced by Hamilton, who called them canonical, an adjective which is used in more than one sense in the literature.

The functional $\mathcal{A}_{H}$ is also called the phase space action (see, for example, [1, p. 37]).

More precisely, assumption (2.5) only guarantees the local invertibility of (2.19). As shown in [20, Sect. 3.11], for (2.21) to be valid in a whole neighborhood, we must strengthen (2.5) by assuming that the mapping

$$ \dot{q}\mapsto \frac{\partial L}{\partial \dot{q}} $$

be a diffeomorphism. In [20], Lagrangians with such a property are also called regular. Dirac considered in [12] a more general form of Hamiltonian dynamics which can also be applied when momenta are not independent functions of (generalized) velocities.

It is straightforward to check that such an $H$ does not depend on $\dot{q}$. Indeed, due to definition (2.19),

$$ \frac{\partial H}{\partial \dot{q}_{i}}= p_{i}- \frac{\partial L}{\partial \dot{q}_{i}}=0. $$

When restricting attention to potential energies that do not depend explicitly on time, the standard qualifier for $f_{U}$ is conservative.

That is, such that $D(q,\cdot )\geqq 0$ for all $q$.

We read in [15]: “The Onsager principle is an extension of Rayleigh’s principle of the least energy dissipation in Stokesian hydrodynamics.”

Perhaps, not anymore, see [56]. For a demonstration of the role of thermal displacement in a virtual-power formulation of thermomechanics, see [55] and [57]. The reader is also referred to [28] for a different theory building on this notion, and to [31], where the theory in [28] is to some extent anticipated.

In [75], when “the general steps for applying OVP [Onsager Variational Principle] to the dynamics of active soft matter” are listed, the first step is “[t]o choose a set of coarse-grained slow variables …to describe the time evolution of the macroscopic state of the system.”

The symmetric matrix whose entries we denoted by $a_{\mathit{ik}}$ is customarily taken positive definite, so that, in particular, all diagonal entries are positive.

This approach characterizes the meaning that we properly attach to analytical thermodynamics; it differs from that implied by Li in the title of his book [44], where analytical is meant to evoke the solid, theoretical structure provided to thermodynamics since the pioneering, elegant work of Gibbs [27] (see p. v of [44]).

This assumption would guarantee that (5.3b) could be reduced to normal form, which is not explicitly contemplated in our theory.

As customary, by this we simply mean that $\mathscr{D}(q_{s};\cdot ,\cdot )\geqq 0$ for all choices of $q_{s}$.

In Truesdell’s words [69, p. 9], “We assume the existence of a second kind of working, $\mathscr{Q}$, called heating, which is not identified with anything from mechanics” (see also [61, p. 125]).

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Title: Analytical Thermodynamics
Authors: Paolo Podio-Guidugli
Epifanio G. Virga
Publication date: 02-03-2023
Publisher: Springer Netherlands
Published in: Journal of Elasticity / Issue 4-5/2023
Print ISSN: 0374-3535
Electronic ISSN: 1573-2681
DOI: https://doi.org/10.1007/s10659-023-09997-6

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