On small local equilibrium systems

1 Introduction

Most work in nonequilibrium thermodynamics is based on a local equilibrium assumption, both in the linear [1] and in the nonlinear regime [2–4]. Within a purely phenomenological approach, the local equilibrium assumption can only be justified by its success. This can be expected when local equilibration times, typically determined by the speed of sound and the subsystem size, are small compared to the time scales for the evolution of the local variables.

The local equilibrium assumption plays an important role in continuum mechanics, say in hydrodynamics or complex fluid mechanics. While macroscopic thermodynamics can be useful even for solving problems in molecular engineering [5], it clearly cannot be applied to infinitesimally small systems and must eventually fail around the nanometer scale. This limit can be pushed by switching from macroscopic thermodynamics to nanothermodynamics [6], which has been developed from Hill’s pioneering theory of thermodynamics for small systems [7].

Strictly speaking, any continuum mechanical equation invoking the assumption of local equilibrium makes sense only in a spatially discretized version, where the characteristic cell volume of the discretization must be sufficiently large to justify the use of macroscopic thermodynamics or at least nanothermodynamics for the local subsystems. This need for discretization is well-known from the discussion of hydrodynamic fluctuations because small systems come with large fluctuations [8, 9]. The purpose of this paper is to discuss the principles and peculiarities of using small system thermodynamics for local equilibrium systems when close to nanometer resolution is required.

For small thermodynamic systems, the Euler equation of macroscopic thermodynamics does not hold. One less equation implies one additional degree of freedom. This additional degree of freedom is the hallmark of small thermodynamic systems. For example, for a discretization of continuum mechanical equations, local thermodynamic equations of state might have an additional explicit dependence on the cell volume. Or, for the transition region between two coexisting phases, there should be one more degree of freedom than expected according to the Gibbs phase rule, if nanothermodynamics is applicable to such regions with significant changes on a molecular length scale at all.

We start with Gibbs’ fundamental form to implement the first and second laws of thermodynamics and to explain intensive and extensive variables; a modified Euler equation is introduced to distinguish small from large systems (Section 2). Legendre–Fenchel transforms are introduced to switch between extensive and intensive variables (Section 3). We then discuss large and small equilibrium systems in terms of variational principles (Sections 4) and illustrate the general ideas for ideal gases (Section 5). Finally, we are ready to discuss the small local equilibrium systems associated with spatial discretizations of hydrodynamics and with interfaces in detail and we obtain some new insights (Section 6). A brief summary and discussion conclude the paper (Section 7).

2 Gibbs’ fundamental form and modified Euler equation

In equilibrium thermodynamics, the following differential form for the exchange of internal energy U of a system with its environment, which is known as Gibbs’ fundamental form, plays a key role,

where each of the d pairs of variables (I _j , X _j ) consists of an intensive variable I _j and an extensive variable X _j . For extensive quantities it is natural to ask “how much of it?” or “how many of them?”, whereas these questions are not meaningful for intensive variables. In other words, extensive variables essentially scale with system size, whereas intensive variables are essentially independent of system size. We make the distinction between extensive and intensive variables even when the thermodynamic limit of infinite system size is not (fully) reached, which motivates the word “essentially” in the previous sentence. In view of the clearly distinct role of the variables I _j and X _j in Gibbs’ fundamental form, such a distinction is meaningful also for small systems. The fundamental form (1) implies that the equations of state for the intensive variables in terms of extensive variables can be obtained as partial derivatives of the thermodynamic potential U(X ₁, …, X _d ) providing all information about the system,

In the limit of large thermodynamic systems, extensive variables scale strictly with system size and intensive variables are strictly independent of system size. Under these homogeneity assumptions, one can proof the Euler equation

For small thermodynamic systems, however, the Euler equation is invalid and we write the modified equation

where the extra energy E is a measure of the smallness of the system. The additional quantity E plays a crucial role in Hill’s discussion of small thermodynamic systems [7].

The fundamental form (1) expresses the first law of thermodynamics. As energy is conserved, the energy U of the system can only be changed via exchange of extensive quantities, such as particles or volume, with the environment. The possibility of heat exchange is resolved as an exchange of entropy, where the second law of thermodynamics establishes the temperature T and the entropy S as the corresponding pair of intensive and extensive variables. This pair is omnipresent in thermodynamics. As the entropy takes care of the degrees of freedom that cannot be controlled macroscopically (“the rest”), we assume (T, S) to be always the last pair in eq. (1), that is, (T, S) = (I _d , X _d ).

Thermodynamic equilibrium states for given values of the extensive variables X ₁, …, X _d are found by minimizing the energy U. To guarantee unique equilibrium states, U(X ₁, …, X _d ) should be a convex function. If, under the assumption of a single phase, the function U is not convex, we must consider the convex envelope of U, which implies that the stable equilibrium state of the system may consist of two or more coexisting phases.

3 Legendre–Fenchel transforms

In practice, one often wishes to replace some of the extensive variables by their intensive partners, most notably, the entropy X _d = S by the more easily measurable and controllable temperature I _d = T, ending up with the Helmholtz free energy as the thermodynamic potential obtained by Legendre transformation. Further replacement of the volume V by the pressure p as an independent variable leads to the Gibbs free energy as the corresponding thermodynamic potential.

Equations (2) and (4) suggest that the extra energy E ( I 1 , … , I d ) is the Legendre transform of U(X ₁, …, X _d ) when all the extensive variables are replaced by their intensive partners as new independent variables. The necessity of U in Eq. (2) being differentiable can be avoided by using the more general Legendre-Fenchel transformation [10],

For differentiable U, the infimum can be evaluated as a minimum characterized by vanishing partial derivatives of the expression in square brackets to recover the standard Legendre transform. The most important features of Legendre–Fenchel transformations are that they transform convex functions into concave functions (and vice versa) and that they are their own inverse transformations. Mathematicians usually prefer a different sign convention that implies the conservation of concavity and convexity under Legendre–Fenchel transformations.

If U is differentiable and we can invert the equations of state (2), we can insert X _j (I ₁, …, I _d ) into eq. (4) to obtain E ( I 1 , … , I d ) . We then find

which suggests that E is the proper thermodynamic potential for the intensive variables. Clearly, this idea cannot work for large thermodynamic systems with E = 0 . This problem arises because the functions I _j (X ₁, …, X _d ) cannot be inverted in standard thermodynamics, as it is impossible to reconstruct extensive variables from strictly intensive ones.

The Euler Eq. (3) and, in view of the fundamental form (1), the equivalent Gibbs–Duhem equation

are consequences of rigorous scaling of extensive variables with system size and rigorous system-size independence of intensive variables. These rigorous properties arise only in the thermodynamic limit. For any finite system, we expect that E is nonzero and can be used as a thermodynamic potential (as we illustrate in Section 5). We can indeed use the fully Legendre–Fenchel transformed quantity E ( I 1 , … , I d ) , which quantifies the violation of the Euler equation, to recognize small systems. Systems with 10²³ degrees of freedom clearly are very close to the thermodynamic limit. If, for a three-dimensional system, we zoom in with a magnification of 10⁶, we already look at rather small thermodynamic systems with 10⁵ degrees of freedom.

4 Variational problems for equilibrium systems

To simplify the further discussion without significant loss of generality, we now switch from the general lists of variables (X ₁, …, X _d ) and (I ₁, …, I _d ) introduced in Eq. (1) to the concrete lists (V, N, S) of extensive and (p, μ, T) of intensive variables for a one-component system, where we have introduced the particle number N and the chemical potential per particle μ in addition to the variables already considered in the beginning of Section 3. According to the preceding section, global equilibrium is characterized by the following variational problem,

If we take the infimum with respect to the extensive variables (V, N, S), a complete list of thermodynamic equations of state is obtained from the derivatives of U(V, N, S), and the infimum is zero according to Eq. (5). We hence obtain the modified Euler Eq. (4) that allows us to calculate E ( p , μ , T ) from U(V, N, S). For large systems, E = 0 ; for small systems, E ( p , μ , T ) is a thermodynamic potential according to Eq. (6).

Taking the infimum in the variational problem (8) with respect to (p, μ, T) as independent variables makes sense only for small systems because, for large systems, the increments of these intensive variables are related by the Gibbs–Duhem Eq. (7). The same thermodynamic equations of state as before are obtained from the derivatives of E ( p , μ , T ) , but in an inverted form. Again, we recover the modified Euler equation, and the infimum takes the value zero.

If we take the infimum in the variational problem (8) with respect to (V, N, T), we still obtain the same set of thermodynamic equations of state, but in a rearranged form. As before, the value of zero for the infimum corresponds to the generalized Euler equation. Even for small systems, the Helmholtz free energy is the appropriate thermodynamic potential because the construction of the proper thermodynamic potential for any choice of the independent variables (one from each pair) is based entirely on the Gibbs fundamental form (1) and Legendre–Fenchel transformation. Similar results are obtained for other mixed triples of extensive and intensive variables.

In order to achieve a more unified description of small and large systems, we pass from extensive variables to intensive densities,

Instead of considering U as a function of V, N, S, we now write

Equation (2) for x ₂ = N and x ₃ = S leads to

whereas Eq. (2) for x ₁ = V implies the modified Euler equation

Legendre transformation from the densities of extensive variables to the intensive variables leads to

Instead of the variational problem (8), we now have the following characterization of global equilibrium,

where the volume V is to be considered as a given constant. Minimization with respect to the densities n, s gives the equations of state (11). Minimization with respect to the intensive variables μ, T gives the equivalent equations of state (13). No problem arises for large systems because the thermodynamic potential for the intensive variables is no longer E , but ɛ − p, which does not vanish for large systems; in other words, the extra energy density ɛ only modifies the thermodynamic potential p. For large systems, u(V, n, s) is independent of V and, according to Eq. (12), ɛ then vanishes. For any choice of two independent variables, the modified Euler Eq. (12) is obtained from the condition that the infimum in the variational problem (14) has the value zero. It is this assumption that allows us to treat V as a constant in the variational problem without losing physically relevant information.

5 Example: ideal gas

In the present section, we illustrate the equations of the previous section for an ideal gas. Whereas, in general, the extra energy E reflects boundary effects and even depends on the shape of a small system, for our toy example, E arises as a purely statistical effect in the (p, μ, T)-ensemble. For discussing this example, we follow Hill’s book [7]. A clear and compact summary of Hill’s approach and some further developments can be found in the more recent book [6].

The functional form of the extra energy E depends on the statistical ensemble used for its evaluation (see table on p. II-102 of [7] or, more generally, [11]). We here use the following result for the (p, μ, T)-ensemble,

with

where k _B is Boltzmann’s constant, h is Planck’s constant, and m the mass of a gas particle. This non-vanishing result for E ( p , μ , T ) can be obtained from the canonical partition function of an ideal gas (in square brackets) as

where summation and integration can be carried out in either order.

From the thermodynamic potential E ( p , μ , T ) we obtain the following equations of state,

and

From the modified Euler equation, we further obtain the caloric equation of state,

Even for a small system containing an ideal gas, we get the usual equations of state (19) and (21) for the single-particle properties pressure and energy. We further note the simple logarithmic relation

so that the extra energy is clearly sub-extensive, and, by solving Eq. (16) for the chemical potential μ,

Note that this expression for the chemical potential has a weak dependence on the particle number N that disappears only in the thermodynamic limit N → ∞. For any finite N, we can reconstruct the particle number N from the essentially intensive variables p, μ, T.

We now switch from the essentially extensive variables N, S to their essentially intensive densities n, s. Motivated by Eq. (13), and based on Eqs. (19) and (22), we write

where Eqs. (16)–(19) imply the following equations for the auxiliary quantity Z = Z(V, μ, T),

The first of this series of equations implies the useful identity

We can now evaluate the derivatives in Eq. (13) to verify some general results,

and

By combining Eqs. (19) and (23), we obtain

With u = (3/2)p and Eq. (28) for eliminating μ in favor of s/n, this expression for p can be rewritten as

For small systems, u(V, n, s) depends explicitly on V, but also on ɛ/u. The latter dependence actually makes this an implicit equation for u. After multiplying by exp{−ɛ/u}, one could expand the exponential on the left-hand side to first order in ɛ/u to get an approximate explicit expression for u.

6 Small local equilibrium systems

As pointed out in the introduction, one often assumes that nonequilibrium systems consist of local equilibrium systems. These local equilibrium systems can be by orders of magnitude smaller than the full nonequilibrium systems, in particular, in the presence of strong inhomogeneities. Such small local equilibrium systems must be expected to be associated with a violation of the Euler equation or, equivalently, with a violation of the Gibbs–Duhem equation. An obvious option for describing these local systems is Hill’s full description of small systems with an additional independent variable representing the deviation from the Euler equation. However, for local equilibrium systems, there is an interesting alternative option: the deviation from the Euler equation could be given by changes compared to the neighborhood around the local equilibrium system. In this section we discuss both options, which can be associated with discretized continuum mechanical equations and with gradient terms for modeling interfaces, respectively.

6.2 Van der Waals’ gradient terms

For small local equilibrium systems, the van der Waals and Cahn-Hilliard theories of interfacial zones [18] suggest an alternative form of Hill’s extra-energy term E characterizing the deviation from the Euler equation: it can depend on gradients of local field variables. This possibility of introducing small-system effects has been elaborated in Chapter 3 of [19] for generalizing the van-der-Waals theory of diffuse interfaces. A more systematic justification of gradient terms has been offered by Langer [20].

For a one-component fluid, one generally characterizes the small local equilibrium systems in an equilibrium interfacial zone between coexisting bulk phases by two thermodynamic variables, where one can use the same variables as for large thermodynamic systems. This remark is less obvious than it might look at first sight because small systems have an extra degree of freedom resulting from the violation of the Euler equation. However, as an equilibrium interfacial zone exists between two coexisting bulk phases, one loses a degree of freedom according to the Gibbs phase rule. In view of the fact that one can use the same number and set of variables through the entire system, including the interfacial zone, one may be tempted to treat equilibrium interfaces like bulk systems with an extra energy density term.

Inspired by the general development in Chapter 3 of [19], we use the particle number density n and the entropy density s as independent variables and let Hill’s extra energy density E depend on the gradients of n and/or s. We consider a weakly nonlocal functional of the form

(32) E [ n , s ] ( r ) = − 1 2 m n n ∂ n ∂ r 2 − 1 2 m s s ∂ s ∂ r 2 − m n s ∂ n ∂ r ⋅ ∂ s ∂ r ,

where the coefficients m _nn , m _ss and m _ns may depend on n( r ) and s( r ). Whereas the violation of the Euler equation expressed by the extra energy density ɛ(V, n, s) defined in Eq. (12) arises from an explicit volume dependence of the thermodynamic equations of state, the small-system behavior given by the extra energy density functional E [ n , s ] ( r ) in Eq. (32) results from the presence of the gradient terms.

In order to find the inhomogeneous equilibrium state of an interface, we need to modify the variational principle (14) by introducing local thermodynamic variables,

(33) inf ∫ u ( n ( r ) , s ( r ) ) − E [ n , s ] ( r ) + p ( μ , T ) − μ n ( r ) − T s ( r ) d 3 r ,

where the integral is over the volume of the entire system. Note that we need to introduce the extra energy density in terms of the variables n, s rather than μ, T because the gradients of the latter variables vanish for equilibrium interfaces. For a full thermodynamic understanding of gradient terms and their proper dependence on various variables, one needs to make a decision about their energetic or entropic origin. By performing the minimization in Eq. (33) for constant coefficients m _nn , m _ss , m _ns (which still may depend on the constant temperature) with respect to n( r ), s( r ), we obtain the following generalization of Eq. (11) (cf. Eq. (3.20) of [19]),

(34) ∂ u ( n , s ) ∂ n − m n n ∂ 2 n ∂ r 2 − m n s ∂ 2 s ∂ r 2 = μ ,

(35) ∂ u ( n , s ) ∂ s − m n s ∂ 2 n ∂ r 2 − m s s ∂ 2 s ∂ r 2 = T .

For equilibrium interfaces, we can choose a fixed temperature T below the critical temperature and the corresponding μ(T) for two-phase coexistence and then solve Eqs. (34) and (35) to obtain the mass and entropy density profiles.

Nonmonotonic profiles of the mass and entropy densities can arise as a consequence of the presence of two independent variables. More general nonlocal functionals E [ n , s ] ( r ) have been used in Eq. (33) for a more realistic modeling of interfaces [19].

If we assume m _ss = m _ns = 0 and constant m _nn , the variational problem (33) can be simplified to

(36) inf ∫ f ( n ( r ) , T ) − E [ n ] ( r ) + p ( μ , T ) − μ n ( r ) d 3 r ,

where f = u − Ts is the local Helmholtz free energy density. Note that f(n, T) + p(μ, T) − μn is the difference between the Helmholtz free energy density and its double tangent at the two coexistent number densities n ₁ and n ₂ at a given temperature T. If we denote this difference by −w(n, t) [19], its typical functional form is shown in Figure 1. For a flat interface, the resulting equation for the number density can be written as

(37) m n n ∂ 2 n ∂ z 2 = − ∂ w ( n , T ) ∂ n ,

where z is the coordinate perpendicular to the interface. Note that this equation possesses the same form as Newton’s equation of motion if n is interpreted as position, z as time, and w as the potential energy. The profile n(z) corresponds to the motion of a particle in the potential w shown in Figure 1 with zero total energy [19]. It is quite remarkable that it is the unstable part of the free energy curve that fully determines the profile n(z).

Figure 1:

The function w(n, T), which is defined as the negative of the difference between the Helmholtz free energy density and its double tangent, as a function of the number density n for a given temperature T in the unstable region between the two coexisting bulk number densities n ₁ and n ₂.

7 Summary and discussion

The Euler and Gibbs–Duhem equations of equilibrium thermodynamics arise in the thermodynamic limit of infinite system size and are hence satisfied for macroscopic thermodynamic systems. Their validity expresses homogeneity properties associated with strictly extensive and intensive variables. For small systems, the extensive and intensive character of thermodynamic variables becomes approximate and the Euler equation is modified by an extra energy term, which implies an additional degree of freedom in small systems. Such small systems arise very naturally from simplifying inhomogeneous equilibrium or nonequilibrium systems by a local equilibrium assumption. For a unified treatment of small and large systems, it is convenient to formulate thermodynamics in terms of a variational principle.

We have considered two particular situations in which small local equilibrium systems can arise: (i) in the numerical solution of hydrodynamic equations by discretization schemes, in which a small cell volume can cause an extra energy that affects thermodynamic relations, and (ii) an extra energy depending on square gradient terms, say in the description of interfaces introduced by van der Waals. For the discussion of the details, we have considered only one-component systems, but the generalization to multi-component systems and systems with additional types of thermodynamic degrees of freedom is straightforward. A number of new insights can be obtained from using the tools of small-systems thermodynamics in these two situations.

In the situation (i), fluctuations become important. Large fluctuations are a hallmark of small systems. Further note that numerical solutions typically require not only spatial discretization, but also a discretization of time. Then the challenging problem of thermodynamic consistency in discrete time arises [21, 22].

Concerning the situation (ii). The corresponding theories typically involve profiles of thermodynamic variables across the interface, requiring that the length scales on which these quantities can be defined are small compared to the molecular thickness of an interface. As the interfacial profiles can be very steep, the existence of thermodynamic variables is implicitly assumed on length scales that can easily be in the sub-nanometer range. Whereas it may be possible to define number or energy densities on such short length scales in a meaningful way, the definition of an entropy density becomes more than questionable. Then also the interpretation of kinetic energy density profiles as temperature profiles may no longer be meaningful because entropy and temperature are deeply related conjugate thermodynamic quantities.

Whereas micro-continuum-mechanical models [23] for interfaces require a questionable super-local equilibrium assumption on sub-nanometer scales, molecular dynamics simulations suggest that, from a thermodynamic perspective, it might be possible to associate a more integral local equilibrium with sharp interfaces [24–30]. There is more to interfaces than highly resolved bulk systems, even though the count of variables might tempt one to believe otherwise (as one degree of freedom gained by the smallness of super-local equilibrium systems is compensated by one degree of freedom lost by the Gibbs phase rule for two coexisting phases).