Search

We give a derivation of a new instantaneous fluctuation relation for an arbitrary phase function which is odd under time reversal. The form of this new relation is not obvious, and involves observing the system along its transient phase space trajectory both before and after the point in time at which the fluctuations are being compared. We demonstrate this relation computationally for a number of phase functions in a shear flow system and show that this non-locality in time is an essential component of the instantaneous fluctuation theorem.

What is now known as the zeroth "law" of thermodynamics was first stated by Maxwell in 1872: at equilibrium, "Bodies whose temperatures are equal to that of the same body have themselves equal temperatures." In the present paper, we give an explicit mathematical proof of the zeroth "law" for classical, deterministic, T-mixing systems. We show that if a body is initially not isothermal it will in the course of time (subject to some simple conditions) relax to isothermal equilibrium where all parts of the system will have the same temperature in accord with the zeroth "law." As part of the derivation we give for the first time, an exact expression for the far from equilibrium thermal conductivity. We also give a general proof that the infinite-time integral, of transient and equilibrium autocorrelation functions of fluxes of non-conserved quantities vanish. This constitutes a proof of what was called the "heat death of the Universe" as was widely discussed in the latter half of the 19th century.

The color-diffusion algorithm is applied to ab initio molecular dynamics simulation of hexagonal LiBH(4) to determine the lithium diffusion coefficient and diffusion mechanisms. Even in the best solid lithium ion conductors, the time scale of ion diffusion is too long to be readily accessible by ab initio molecular dynamics at a reasonable computational cost. In our nonequilibrium method, rare events are accelerated by the application of an artificial external field acting on the mobile species; the system response to this perturbation is accurately described in the framework of linear response theory and is directly related to the diffusion coefficient, thus resulting in a controllable approximation. The calculated lithium ionic conductivity of LiBH(4) closely matches published measurements, and the diffusion mechanism can be elucidated directly from the generated trajectory.

Relaxation of a system to equilibrium is as ubiquitous, essential, and as poorly quantified as any phenomena in physics. For over a century, the most precise description of relaxation has been Boltzmann's H-theorem, predicting that a uniform ideal gas will relax monotonically. Recently, the relaxation theorem has shown that the approach to equilibrium can be quantified in terms of the dissipation function first defined in the proof of the Evans-Searles fluctuation theorem. Here, we provide the first demonstration of the relaxation theorem through simulation of a simple fluid system that generates a non-monotonic relaxation to equilibrium.

In this paper, we re-visit Gibbs' second (unresolved) paradox, namely the constancy of the fine-grained Gibbs entropy for autonomous Hamiltonian systems. We compare and contrast the different roles played by dissipation and entropy both at equilibrium where dissipation is identically zero and away from equilibrium where entropy cannot be defined and seems unnecessary in any case. Away from equilibrium dissipation is a powerful quantity that can always be defined and that appears as the central argument of numerous exact theorems: the fluctuation, relaxation, and dissipation theorems and the newly derived Clausius inequality.

In 1854 Clausius proved the famous theorem that bears his name by assuming the second "law" of thermodynamics. In the present paper we give a proof that requires no such assumption. Our proof rests on the laws of mechanics, a T-mixing property, an ergodic consistency condition, and on the axiom of causality. Our result relies on some recently derived theorems, such as the Evans-Searles and the Crooks fluctuation theorems and the recently discovered relaxation and dissipation theorems., (© 2011 American Institute of Physics)

In 2005, Bright et al. gave numerical evidence that among the family of time reversible deterministic thermostats known as μ-thermostats, the conventional μ=1 thermostat proposed by Hoover and Evans is the only thermostat that is capable of generating an equilibrium state. Using the recently discovered relaxation theorem, we give a mathematical proof that this is true.

In the past decade there has been considerable interest in nonequilibrium free energy theorems. These theorems show how the difference between the free energies of two equilibrium states of a system are related to distributions of work along nonequilibrium pathways. Here we significantly extend these theorems by relating the possibly inaccessible equilibrium average of any phase variable to the distribution of work along nonequilibrium paths. The result is elegant and analogous to the static umbrella sampling used with Monte Carlo simulations. The umbrella weight, which is related to the nonequilibrium work, is easy to obtain and efficient to compute. Furthermore, its average may be obtained from the existing nonequilibrium free energy theorem.

It has recently become apparent that the dissipation function, first defined by Evans and Searles [J. Chem. Phys. 113, 3503 (2000)], is one of the most important functions in classical nonequilibrium statistical mechanics. It is the argument of the Evans-Searles fluctuation theorem, the dissipation theorem, and the relaxation theorems. It is a function of both the initial distribution and the dynamics. We pose the following question: How does the dissipation function change if we define that function with respect to the time evolving phase space distribution as one relaxes from the initial equilibrium distribution toward the nonequilibrium steady state distribution? We prove that this covariant dissipation function has a rather simple fixed relationship to the dissipation function defined with respect to the initial distribution function. We also show that there is no exact, time-local, Evans-Searles nonequilibrium steady state fluctuation relation for deterministic systems. Only an asymptotic version exists.

We study the statistical mechanics of thermal conduction in a classical many-body system that is in contact with two thermal reservoirs maintained at different temperatures. The ratio of the probabilities, that when observed for a finite time, the time averaged heat flux flows in and against the direction required by Fourier's Law for heat flow, is derived from first principles. This result is obtained using the transient fluctuation theorem. We show that the argument of that theorem, namely, the dissipation function is, close to equilibrium, equal to a microscopic expression for the entropy production. We also prove that if transient time correlation functions of smooth zero mean variables decay to zero at long times, the system will relax to a unique nonequilibrium steady state, and for this state, the thermal conductivity must be positive. Our expressions are tested using nonequilibrium molecular dynamics simulations of heat flow between thermostated walls.

We examine the question of whether fluids and crystals are differentiated on the basis of their zero frequency shear moduli or their limiting zero frequency shear viscosity. We show that while fluids, in contrast with crystals, do have a zero value for their shear modulus, in contradiction to a widespread presumption, a crystal does not have an infinite or exceedingly large value for its limiting zero frequency shear viscosity. In fact, while the limiting shear viscosity of a crystal is much larger than that of the liquid from which it is formed, its viscosity is much less than that of the corresponding glass that may form assuming the liquid is a good enough glass former.

The mathematics of time-dependent nonlinear response theory frequently leads to results which although formally exact, are not amenable to experimental application or even to use in computer simulations. Here we give a rigorous derivation of a tractable expression for the thermostatted nonlinear response of classical many body systems to time-dependent dissipative fields. The theory also allows for the concurrent parametric transformation of the system Hamiltonian. Our analysis shows once again the intimate relationship between nonequilibrium free-energy relations such as the Jarzynski equality and nonlinear response theory. We make a few remarks concerning the maximum entropy production approximation.

The Jarzynski equality and the Crooks fluctuation theorem enable the calculation of the change in a system's free energy from nonequilibrium path integrals. These relations consider processes where the system is driven out of equilibrium by a mechanical external agent while remaining in contact with a thermal reservoir at a fixed temperature. We generalize these relations to describe processes driven by any type of external agent, be it thermal or mechanical. Attention is given to the case of a system, initially in equilibrium, that is driven through a temperature change by a heat reservoir. The results are cast in a form applicable to experiments.

Recently, there has been considerable interest in the fluctuation theorem (FT). The Evans-Searles FT shows how time reversible microscopic dynamics leads to irreversible macroscopic behavior as the system size or observation time increases. We show that the argument of this FT, the dissipation function, plays a central role in nonlinear response theory and derive the dissipation theorem, giving exact relations for nonlinear response of classical N-body systems that are more widely applicable than previous expressions. These expressions should be verifiable experimentally. When linearized they reduce to the well-known Green-Kubo expressions for linear response.

We examine the question of whether the formal expressions of equilibrium statistical mechanics can be applied to time independent nondissipative systems that are not in true thermodynamic equilibrium and are nonergodic. By assuming that the phase space may be divided into time independent, locally ergodic domains, we argue that within such domains the relative probabilities of microstates are given by the standard Boltzmann weights. In contrast to previous energy landscape treatments that have been developed specifically for the glass transition, we do not impose an a priori knowledge of the interdomain population distribution. Assuming that these domains are robust with respect to small changes in thermodynamic state variables we derive a variety of fluctuation formulas for these systems. We verify our theoretical results using molecular dynamics simulations on a model glass forming system. Nonequilibrium transient fluctuation relations are derived for the fluctuations resulting from a sudden finite change to the system's temperature or pressure and these are shown to be consistent with the simulation results. The necessary and sufficient conditions for these relations to be valid are that the domains are internally populated by Boltzmann statistics and that the domains are robust. The transient fluctuation relations thus provide an independent quantitative justification for the assumptions used in our statistical mechanical treatment of these systems.

Molecular dynamics simulations are performed on a realistic glass forming model system. The linear and nonlinear response domains are explored numerically for the case where one of the particles interacts with a constant external force. As the temperature is lowered towards the glass transition, we find that the range of fields over which the response is linear shrinks towards zero. We show that the time required for convergence of the steady state fluctuation theorem and the valid application of the central limit theorem becomes very large as the glass transition is approached. This in turn implies that the domain over which a linear response can be observed becomes progressively smaller.

Deterministic thermostats are frequently employed in nonequilibrium molecular dynamics simulations in order to remove the heat produced irreversibly over the course of such simulations. The simplest thermostat is the Gaussian thermostat, which satisfies Gauss's principle of least constraint and fixes the peculiar kinetic energy. There are of course infinitely many ways to thermostat systems, e.g., by fixing sigma(i)/p(i)/mu+l. In the present paper we provide, for the first time, convincing arguments as to why the conventional Gaussian isokinetic thermostat (mu = 1) is unique in this class. We show that this thermostat minimizes the phase space compression and is the only thermostat for which the conjugate pairing rule holds. Moreover, it is shown that for finite sized systems in the absence of an applied dissipative field, all other thermostats (mu not = 1) perform work on the system in the same manner as a dissipative field while simultaneously removing the dissipative heat so generated. All other thermostats (mu not = 1) are thus autodissipative. Among all mu, thermostats, only the mu = 1 Gaussian thermostat permits an equilibrium state.

The fluctuation relation of the Gallavotti-Cohen fluctuation theorem (GCFT) concerns fluctuations in the phase-space compression rate of dissipative, reversible dynamical systems. It has been proven for Anosov systems, but it is expected to apply more generally. This raises the question of which non-Anosov systems satisfy the fluctuation relation. We analyze time-dependent fluctuations in the phase space compression rate of a class of N-particle systems that are at equilibrium or in near equilibrium steady states. This class does not include Anosov systems or isoenergetic systems; however, it includes most steady-state systems considered in molecular-dynamics simulations of realistic systems. We argue that the fluctuations of the phase-space compression rate of these systems at or near equilibrium do not satisfy the fluctuation relation of the GCFT, although the discrepancies become somewhat smaller as the systems move further from equilibrium. In contrast, similar fluctuation relations for an appropriately defined dissipation function appear to hold both near and far from equilibrium.