1 Introduction

Phenomenological evolution equations with a thermodynamic structure can be enhanced by adding fluctuations. These fluctuations represent the effect of more microscopic, fast degrees of freedom that have been neglected in the phenomenological equations. Whether the fluctuations cause only small corrections or have far-reaching consequences depends on the particular problem of interest.

A simple but important example for the benefits of fluctuation enhancement is provided by the motion of small particles suspended in fluids. If the suspended particles are sufficiently small, such as pollen particles in water, one observes a wild random motion of the particles, which is the famous Brownian motion resulting from incessant collisions with small molecules in the fluid. If the fluid is described by the phenomenological hydrodynamic equations, which specify the evolution of density, velocity and temperature fields, Brownian motion cannot be explained. By adding fluctuations to phenomenological hydrodynamics, one incorporates important features of the molecular nature of the fluid and Brownian motion becomes accessible. An important experimental technique known as microbead rheology (see, for example, Chapter 25 of the textbook [1] and references therein) is actually based on the observation of the random motion of small particles in complex fluids, from which the viscoelastic fluid properties can be inferred.

An even more striking example for the importance of fluctuations is dynamic light scattering by a low-molecular-weight liquid, say water (see, for example, the monograph [2] or Chapter 26 of the textbook [1]). This experimental technique is sensitive to time-dependent correlations in mass density on the length scale of the wavelength of light. Typical length scales are below 10−7m, typical time scales can go down to 10−10s, where these two scales are related by the speed of sound. Ultimately, it is the polarizability of molecules that matters for the scattering of electromagnetic waves. Information on such short length scales is not available in the phenomenological hydrodynamic equations. By adding fluctuations to hydrodynamics, we obtain precisely the small-scale information about density correlations that is required to describe dynamic light scattering [3]. This example demonstrates the usefulness of fluctuating hydrodynamics impressively.

The standard reference for fluctuating hydrodynamics is the textbook by Landau and Lifshitz [4]. To respect conservation laws, Gaussian fluctuations are introduced for the fluxes of conserved quantities rather than the conserved quantities themselves. At any time, the local-equilibrium fluctuations are characterized by Einstein’s fluctuation theory obtained from equilibrium statistical mechanics (see, for example, Section 10.B of [5]). According to Onsager’s regression hypothesis [6], [7], the decay of fluctuations is governed by the phenomenological hydrodynamic equations.

In the context of numerical solutions of hydrodynamic equations, the relevance of fluctuations in a large number of situations has been compiled and discussed by Donev and coworkers. Diffusive transport is strongly enhanced by thermal velocity fluctuations [8]. Fluctuations also have a strong impact on the spinodal decomposition of multicomponent systems [9]. Moreover, fluctuations in reactive systems have a strong effect on giant long-range correlated concentration fluctuations, accelerate pattern formation in spatially homogeneous systems and lead to a qualitatively different disordered pattern behind a traveling wave [10], [11].

Can hydrodynamic fluctuations in multicomponent systems be described in the same way as those in one-component systems? If we consider only the resulting diffusion effects, the answer is yes. If we moreover allow for chemical reactions between the different components, the answer is no [12]. In this paper, building upon the works [13], [14], we first show how different types of noise correspond to different thermodynamic structures (force-flux constitutive laws) of the phenomenological equations for isolated systems. Then, as our central contribution, we present the consequences of this correspondence for nonequilibrium statistical mechanics, by answering the following two questions. (i) Given a microscopic model and a meaningful coarse-graining map, how can one find the thermodynamic structure of the macroscopic, phenomenological equation? (ii) In the opposite direction, given a phenomenological equation and its thermodynamic structure, how can we add fluctuations? For these procedures we use the terms “coarse-graining” and “noise enhancement.”

This paper pursues the following line of thought. We start from a class of phenomenological evolution equations for isolated systems with a particular thermodynamic structure (Section 2). To enhance these phenomenological equations, we do not restrict ourselves to continuous Gaussian noise but rather allow for general Markov processes, including jump processes. We then try to recognize the proper types of noise to be used for enhancing phenomenological equations in the thermodynamic structure of those equations. This attempt leads us to the very general fluctuation-dissipation theorem that is at the heart of this paper (Section 3). Finally, we analyze the noise resulting from more microscopic descriptions to extract the detailed form of more macroscopic, phenomenological evolution equations, which is a key task of nonequilibrium statistical mechanics (Section 4).

For the theory of systems in thermodynamic equilibrium, which do not macroscopically evolve in time, we know that the general recipes for calculating thermodynamic potentials by statistical mechanics have not yet been derived in any rigorous manner: for instance, there is no definite proof that the thermodynamic entropy, which is a macroscopically measurable quantity [15], is the same entropy computed by statistical mechanics (for an attempt in this direction, see [16]). Nevertheless, equilibrium statistical mechanics has been formulated in such an elegant and convincing way by Gibbs, whose powerful formulation is based on ideas of Boltzmann and Maxwell, that it is now generally accepted and used with greatest confidence. For nonequilibrium systems, we certainly cannot expect anything better than compelling heuristic arguments and an elegant mathematical formulation of the recipes that can be used to extract the thermodynamic structure of phenomenological evolution equations from microscopic dynamics. In particular, the announced generalized fluctuation-dissipation theorem does not have the status of a mathematical theorem, but rather of a plausible postulate or axiom.

The main goal of all our efforts is to provide the tools for condensing experimental observations, partial understanding, statistical arguments, and intuition into a consistent multiscale description for a given problem of interest. For example, available background knowledge should be used to guide the focus of statistical mechanics and to make its procedures more efficient, and intuition about molecular processes should guide appropriate phenomenological modeling even if systematic statistical mechanics is prohibitively expensive. We strive for a multiscale approach with reassuring mutual consistency between different levels of description.

2 A framework for nonequilibrium thermodynamics

To carry out the program sketched in the introduction, it is essential to build on a sound framework for nonequilibrium thermodynamics. On the one hand, if we wish to enhance a phenomenological equation by adding fluctuations, we must be sure that the phenomenological equation is thermodynamically consistent so that a recipe for introducing thermal fluctuations can make physical sense. On the other hand, a major goal of statistical mechanics is to evaluate expressions for the building blocks of a thermodynamic description from a more microscopic description. At equilibrium, we need to find a single thermodynamic potential that characterizes all the thermodynamic equations of state of the system. Nonequilibrium thermodynamics requires more.

Our discussion is based on the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) formulation of time-evolution equations for nonequilibrium isolated systems [17], [18], [19], [20], where two different formulations of the fundamental time evolution have already been offered in eqs. (4) and (13) of the original paper [17]. Both formulations provide autonomous evolution equations for a list of independent variables x that describe an isolated nonequilibrium system of interest, and both formulations rely on an additive superposition of reversible and irreversible contributions to dynamics. The fact that the GENERIC framework deals with isolated systems represents no essential limitation, since the equations for local field theories, the most frequent models for real systems, are independent of boundary conditions; in other cases, it may be necessary to include the environment in the description. Attempts to generalize the GENERIC framework to open systems have been made in [21], [22], [23].

For simplicity, here we assume that the list of variables x is finite, x∈Rd. The first formulation of GENERIC in [17] is given by the equation

The quantities E and S are the total energy and entropy as functions of the system variables x, and L and M are certain linear operators, or matrices, which are also allowed to depend on x. The two contributions to the time evolution of x generated by the energy E and the entropy S in eq. (1) are the reversible and irreversible contributions, respectively. We refer to L and M as the Poisson and friction matrices.

The GENERIC framework postulates a number of properties for the four building blocks E, S, L and M, most of which are related to physically sound balance laws for energy and entropy (which, in turn, are associated with the fundamental laws of thermodynamics). The practical advantage of restricting ourselves to isolated systems is that conservation laws can be formulated more easily. The conservation of energy by reversible dynamics is guaranteed by requiring the Poisson matrix L to be antisymmetric. The conservation of energy by irreversible dynamics is assumed for all choices of the generator S; this assumption means that ∂E/∂x is a left eigenvector of the matrix M with eigenvalue zero, thus implying degeneracy of the friction matrix M. The conservation of entropy by reversible dynamics, which actually is a hallmark of reversibility, is assumed for all choices of the generator E; this assumption means that ∂S/∂x is an eigenvector of the matrix L with eigenvalue zero, thus implying degeneracy of the Poisson matrix L, too. Finally, the friction matrix M is assumed to be positive-semidefinite. This assumption is a very strong formulation of the second law of thermodynamics because it implies that the irreversible contribution to the rate of change of S, that is, dS/dt, is nonnegative for any choice of S, not just for the physical entropy.

There are further properties of L and M that are not related to the balance laws for energy and entropy. While the Poisson matrix L has to be antisymmetric, the symmetry properties of the friction matrix M are less obvious. Often M is assumed to be symmetric, but requiring Onsager–Casimir symmetry is more appropriate (see Sections 3.2.1 and 7.2.4 of [19] as well as [24]). A highly restrictive condition on the matrix L is given by the Jacobi identity for the Poisson bracket associated with L, which is defined by {A,B}=(∂A/∂x)·L(∂B/∂x). This property expresses the time-structure invariance of reversible dynamics [25]. The version (1) of GENERIC is also known as a metriplectic structure [26]. Note, however, that the “metric” M(x) is degenerate and may be nonsymmetric (for example, in modeling turbulence [24] or slip [27]).

The second formulation of GENERIC in [17] is given by the equation

In eq. (2), the dissipation potential Ψ∗ is a convex real-valued function of ξ that has its minimum at 0, where Ψ∗(x,0)=0 (for the origins of using dissipation potentials in irreversible thermodynamics, see [28], [29], [30], [31], [32]; see also the remarks in Section 2.9 of [33]). The potential Ψ∗ can have an additional explicit dependence on x. The formulation of irreversible dynamics in terms of a dissipation potential is also known as generalized gradient flow (for details, see [13] and references therein). In the following, we hence distinguish between the GENERIC implementation of irreversible dynamics by gradient flows in eq. (1) and by generalized gradient flows in eq. (2); equivalently, we refer to gradient flows based on friction matrices and based on dissipation potentials. For symmetric M(x), the quadratic dissipation potential Ψ∗(x,ξ)=(1/2)ξ·M(x)ξ reproduces the gradient flow appearing in eq. (1).

The respective advantages of the formulations (1) and (2) have been discussed in the literature, in most detail in [34]. For the formulation of irreversible dynamics based on dissipation potentials, the strong formulation of energy conservation and problems associated with dimensional arguments of inhomogeneous functions have been addressed in [35], where, based on physical arguments, also the combination of the formulations (1) and (2) has been advocated and elaborated.

A preference for the formulation (1) or (2) has often been considered as a matter of taste. Only eq. (1) can handle Casimir symmetry, whereas eq. (2) is clearly more natural for chemical reactions and for collisions in the Boltzmann equation. In the present paper, we show that the choice of a formulation actually implies the type of noise by which a phenomenological equation can be enhanced. The choice should be based on the physical situation and hence be made with deliberation.

3 Fluctuation-dissipation theorem

The fluctuation-dissipation theorem plays a key role in describing nonequilibrium systems. As a general principle, it was first formulated by Nyquist [36] in 1928 and later derived by Callen and Welton [37], but important special cases had been noted in the preceding decades. Kubo and coworkers [38] proposed a scheme for classifying the various types of formulas referred to as fluctuation-dissipation relations.

Here we consider the fluctuation-dissipation theorem as the principle that establishes a relationship between the (generalized) gradient flows characterizing the irreversible contribution to GENERIC on the one hand and the appropriate fluctuation enhancement of these thermodynamically structured equations on the other hand. The Markov processes providing the fluctuation enhancement can be characterized in various ways: by stochastic differential equations, transition rates, functional integrals, infinitesimal generators, or nonlinear generators.

4 Recipes for nonequilibrium statistical mechanics

In the context of GENERIC with friction matrices or fluctuations modeled by diffusion processes, the four building blocks characterizing the structure of the phenomenological equations can be obtained by statistical mechanics as elaborated in Chapter 6 of [19] or in [65]. The generators energy E and entropy S of reversible and irreversible dynamics, respectively, as well as the Poisson matrix L, require only static information that can be obtained most efficiently from Monte Carlo simulations of nonequilibrium ensembles. Dynamic simulations are required for evaluating the friction matrix M. In view of the cumulant generating function (29), we can extract M from

In practice, the left-hand side is evaluated for small time increments τ and, for this reason, we may just compute

since the term τ⟨v⟩xτ⟨vT⟩xτ is of order τ1, whereas τvvTxτ is of order τ0. In a simulation, one creates an ensemble of microscopic initial conditions that are consistent with the thermodynamic variables x and that are evolved with the microscopic dynamics. This procedure for evaluating second moments clarifies what we mean by the “noise idealization of fast dynamics” and by “fluctuation-enhanced phenomenological dynamics.” As discussed in the justification of the method of steepest descent in the paragraph before eq. (26), τ should be small from the macroscopic perspective, but large from a microscopic perspective. More precisely, for diffusion processes, this means that τ should be the separating time scale between the fast processes idealized as noise and the slow processes described by thermodynamic evolution equations.

Dynamic simulations are performed only over the intermediate time scale τ, not over macroscopic times. This is where the efficiency of thermodynamically guided simulations comes from [65]. If the friction matrix M is independent of x, it is convenient to average over an ensemble of initial conditions for different x, for example, an equilibrium ensemble. The above procedure for diffusion processes is closely related to the more common approach based on Green–Kubo formulas [38], [66].

Statistical mechanics for GENERIC with dissipation potential or fluctuations modeled by general Markov processes is based on eq. (28). For a given x, one needs to choose α and τ. The choice of α is determined by the range of ξ values of interest according to eq. (27). The intermediate time scale τ should be such that a small but nonnegligible fraction of the simulated trajectories contains a jump during the period τ.

The left-hand side of eq. (28) can be evaluated in exactly the same way as the second moment in the Green–Kubo formula (31). A fit to the right-hand side provides the noise idealization for the neglected degrees of freedom and the resulting dissipation potential according to the generalized fluctuation-dissipation theorem. If the result is plotted for fixed x as a function of ξ, according to eq. (28), we obtain the dissipation potential Ψ∗(x,ξ), except for constant and linear contributions. For ξ=∂S/∂x, the right-hand side is zero; for ξ=0, the right-hand side is −Ψ∗(x,∂S/∂x). The static building blocks E, S and L can be extracted from molecular simulations exactly as described before for GENERIC based on friction matrices.

5 Summary and outlook

The main result of the present paper is a very general fluctuation-dissipation theorem that relates the dissipative contribution in a thermodynamically consistent evolution equation to a fluctuation enhancement of that equation in terms of a Markov process. In its most general form, dissipative dynamics is defined in terms of a dissipation potential Ψ∗ in eq. (2). The Markov process providing the fluctuation enhancement of the phenomenological eq. (2) can be characterized in several equivalent ways (under suitable assumptions):

1. by the probability distribution (9) in the space of trajectories, where the general Lagrangian F is given in eq. (18);

2. by the infinitesimal generator (20) associated with the transition probabilities pxτ given in eq. (21) or eq. (23), where the correction (22) should be applied;

3. by a nonlinear infinitesimal generator obtained as a limiting case of eq. (28) for small time steps τ (compared to macroscopic time scales).

The fact that fluctuations and dissipation are two sides of the same coin can be exploited in different ways which, when properly combined, lead to a consistent multiscale description of nonequilibrium systems. On the one hand, phenomenological evolution equations rooted in nonequilibrium thermodynamics can be enhanced by introducing fluctuations representing the essential effects of neglected microscopic degrees of freedom. Important examples include the hydrodynamic equations for the discussion of dynamic light scattering and microbead rheology. On the other hand, by analyzing the fluctuations of microscopic simulations, one obtains recipes for extracting the dynamic material information expressed through dissipation potentials. The general recipe of nonequilibrium statistical mechanics is contained in eq. (28). This equation unifies and generalizes some famous results: the Green–Kubo formulas for transport coefficients (associated with gradient flows, friction matrices and continuous noise) and the Kramers formula for chemical reactions rates (associated with generalized gradient flows, dissipation potentials and jump processes).

In the subsequent paper, we illustrate how the general recipes of nonequilibrium statistical mechanics can be applied to obtain the dissipation potential for purely dissipative systems and, in particular, for a simple chemical reaction. Applications of greater practical relevance will require the development of sophisticated importance-sampling techniques.