Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: I. Generalized fluctuation-dissipation theorem

The paper is devoted to the theory of thermal fluctuations in nonlinear macroscopic systems and to the derivation of variational principles of nonlinear nonequilibrium thermodynamics. In the first part of the paper rigorous universal fluctuation-dissipation relations for nonlinear classical and quantum systems, subjected to dynamic as well as thermodynamic perturbations, are derived and analyzed. General expressions for dissipative fluxes and nonlinear transfer coefficients with the help of fluctuation cumulants are found. The canonical structure of nonlinear evolution equations of macrovariables is derived and the rule of introducing langevinian random forces into these equations, in accordance with fluctuation-dissipation relations. A Markovian theory of fluctuations in a stationary nonequilibrium state is constructed.     

Plasma response functions,fluctuation-dissipation relations and the velocity-average-approximation

We derive and list relationships for quadratic and cubic static response functions and connect them with three- and four-point functions through generalized fluctuation-dissipation relations. We also introduce the useful concept of “response function of the second kind.” to describe the reponse of a system through the perturbation of its two-point functions. Next, we point out that the VAA (velocity-average-approximation) introduced earlier as a part of a dynamical approximation scheme for strongly coupled one-component plasmas is exact in the static limit. This observation, combined with the fluctuation-dissipation relations, allows one to derive a hierarchy of equations for response functions of increasing nonlinearity, which is equivalent to the BGY hierarchy for particle distribution functions.     

Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: II. Kinetic potential and variational principles for nonlinear irreversible processes

On the basis of a complete system of fluctuation-dissipation relations, considered in the first part of this series, a variational principle for nonlinear irreversible processes is derived. According to this principle the virtual entropy production functional (analogous to the action in mechanics) has an absolute minimum meaning on the real trajectory of a system. The universal structure of the “kinetic potential” and the “lagrangian” of a system, each contain complete information about fluctuations of macrovariables. The connection of the lagrangian with the markovian kinetic operator of macrovariables is stated. Fundamental properties of dissipative potentials, reflecting microscopic reversibility, are considered. The derived variational principle can be applied to closed systems (the steady state of which is equilibrium) as well as to open ones (when external dynamic forces cause entropy flux through the system and put it into a steady non-equilibrium state). Canonical transformations of macrovariables are considered.     

On the theory of nonequilibrium s-particle distribution functions

A study is made of systems with a small departure from an equilibrium state. It is shown that the transport coefficients that characterize the response of the system to a mechanical perturbation can be expressed in terms of the equilibrium s-particle distribution functions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 7, pp. 63–67, July, 1971.     

On the fluctuation-dissipation theorem in a disordered nonequilibrium system

We present a modified perturbational calculation for a model with quenched disorder in nonequilibrium. A systematic way is proposed to obtain corrections to the usual fluctuation-dissipation theorem.     

Observable dependence of fluctuation-dissipation relations and effective temperatures

We study the nonequilibrium fluctuation-dissipation theorem (FDT) in the glass phase of Bouchaud's trap model. We incorporate an arbitrary observable m and obtain its correlation and response functions in closed form. A limiting nonequilibrium FDT plot is approached at long times for most choices of m. In contrast to standard mean field models, however, the shape of the plot depends nontrivially on the observable, and its slope varies continuously even though there is a single scaling of relaxation times with age. Nonequilibrium FDT plots can therefore not be used to define a meaningful effective temperature T(eff) in this model. Consequences for the wider applicability of an FDT-derived T(eff) are discussed.     

Derivation of the linear and nonlinear non-Markov fluctuation-dissipation relations of the first kind for a dynamical model

A dynamical system consisting of a subsystem having the variablesz=(q,p) and of another dynamical system (thermostat) is considered in the nonquantum case. Using a dynamical equation, it is shown that the linear and quadratic non-Markov fluctuation-dissipation relations (FDRs) of the first kind are valid in the first nonvanishing approximation in interaction constants. Applying these FDRs, one can determine the statistical properties of the fluctuations when the form of the nonlinear phenomenological equation is known. The non-Markov FDRs of the first kind are the direct generalization (to the inertial case) of the Markov FDRs that are the consequence of detailed balance.     

Generalized off-equilibrium fluctuation-dissipation relations in random Ising systems

We show that the numerical method based on the off-equilibrium fluctuation-dissipation relation does work and is very useful and powerful in the study of disordered systems which show a very slow dynamics. We have verified that it gives the right information in the known cases (diluted ferromagnets and random field Ising model far from the critical point) and we used it to obtain more convincing results on the frozen phase of four-dimensional spin glasses. Moreover we used it to study the Griffiths phase of the diluted and the random field Ising models. Received 1 December 1998 and Received in final form 17 February 1999     

Calculation of the fluctuation-dissipation ratio for the nonequilibrium critical behavior of disordered systems

The values of a new universal parameter characterizing a nonequilibrium critical behavior, namely, the fluctuation-dissipation ratio specifying a fundamental relation between the dynamic response function and the correlation function, are calculated for the disordered three-dimensional Ising model. The analysis of the two-time dependence for autocorrelation functions and the ac susceptibility for the systems with spin densities p = 1.0, 0.8, and 0.6 shows the aging effects characterized by the anomalous slowing of relaxation in the system with the growth of the waiting time and the violation of the fluctuation-dissipation theorem. To improve the accuracy of the ac susceptibility calculations, the “thermal bath” technique has been used without introducing the applied magnetic field in the simulation. It has been shown that the structural defects lead to the pronounced enhancement of the aging effects.     

Portfolio theory of optimal isometric force production: Variability predictions and nonequilibrium fluctuation-dissipation theorem

We derive a fundamental relationship between the mean and the variability of isometric force. The relationship arises from an optimal collection of active motor units such that the force variability assumes a minimum (optimal isometric force). The relationship is shown to be independent of the explicit motor unit properties and of the dynamical features of isometric force production. A constant coefficient of variation in the asymptotic regime and a nonequilibrium fluctuation-dissipation theorem for optimal isometric force are predicted.     

Recurrence relations for the expansion of partial distribution functions

An equation is examined for the generating functional of partial distribution functions. The solution of this equation can be written as an integropower series whose kernels determine all the distribution functions. A recurrence relation is obtained which can be used to construct successively all the kernels. This expansion is the same as an expansion in terms of irreducible coupled diagrams.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 12, No. 7, pp. 104–108, July, 1969.     

Establishing experiments to find material functions in tensor nonlinear constitutive relations

The basic scheme of an establishing experiment to find two material functions of two invariants in the constitutive relations of a nonlinear incompressible tensor medium, is described. A combination of radial spreading and two one-dimensional shears in mutually perpendicular directions observed in a cylindrical layer is taken as the base flow of such a medium. Percolation is possible at the boundaries of cylinders, but the so-called tangential component of the medium coincides with the velocity of cylinder motion. The fundamental question of a viscous potential existing for each medium is posed.     

Stochastic theory of nonlinear nonequilibrium processes

Arguments about the stochastic nature of the changes in thermodynamic parameters are used to construct a nonlinear version of the nonequilibrium thermodynamics of scalar processes, with account taken of crossover effects. The formalism is generalized to the case of vector processes. A cyclic chemical reaction in an ideal gas is treated as an example.Translated from Izvestiya Vyssikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 55–63, September, 1970.     

De Broglie-type relations from nonlinear evolution equations

De Broglie-type relations for the energy and momentum of the space-localized solutions to a class of nonlinear complex Hamiltonian evolution equations are derived without any “external assumptions.” It is found that the Hermitian norm of the same solutions plays a more fundamental role than a mere normalization constant. The quantum commutator is obtained from the infinite-dimensional complex Poisson bracket, and it is in complete agreement with the above find. The significance that such relationships should exist, while being entirely independent of the concepts of pointlike particle and/or point charge, is briefly discussed.     

Order relations for lattice sums from order relations for theta functions

Let {τ} and {γ} denote mutually reciprocal unit Bravais lattices in an n-dimensional Euclidean space, and consider the Theta Functions (TF's) $\text{V}{}_{\mathrm{\tau }}\text{(t) = t}{}^{\mathrm{<mtext>n</mtext><mtext>4</mtext>}}\text{∑}{}_{\mathrm{\tau }}\text{exp}\text{(?πtτ}{}^2\text{)}$ for all 0 < t < ∞. By showing how to evaluate a larger class of sums $\text{Z}{}_{\mathrm{\tau }}{}^{\mathrm{(K)}}\text{(t)  π}{}^{\mathrm{k}}\text{t}{}^{\mathrm{<mtext>k\; +\; </mtext><mtext>n</mtext><mtext>4</mtext>}}\text{∑}{}_{\mathrm{\tau }}{}_{\mathrm{r}}{}^{\mathrm{2k}}\text{exp (?πtτ}{}^2\text{)}$, k a nonnegative integer, we are able to evaluate any derivative of the V-functions. With this information we find order relations for the TF's on the cubic lattices in three dimensions. Coupling these relations with Ewald's Theta Function method, we secure order relations for Lennard-Jones, Chaba-Pathria, and other lattice sums on cubic lattices. We also sketch extensions to non-Bravais lattices and give an order relation for TF's on the non-Bravais hexagonal closepacked and the Bravais facecentered cubic.     

On the nonequilibrium statistical mechanics of a binary mixture. I. The distribution functions

In this paper, we obtain the generalization of the BBGKY hierarchy for a binary mixture of chemically neutral particles. Using modified boundary conditions different from the ones proposed by Bogoliubov, we solve the hierarchy, and obtain explicitly the set of two-particle distribution functions for the several species of the mixture, up to first order in the density.     

Microscopic description of aging dynamics: fluctuation-dissipation relations,effective temperature,and heterogeneities

We consider the dynamics of a diluted mean-field spin glass model in the aging regime. The model presents a particularly rich heterogeneous behavior. In order to catch this behavior, we perform a spin-by-spin analysis for a given disorder realization. We confirm the connection between statics and dynamics at the level of single degrees of freedom. Moreover, working with single-site quantities, we can introduce a new response-vs-correlation plot, which clearly shows how heterogeneous degrees of freedom undergo coherent structural rearrangements. We discuss the general scenario which emerges from our work and (possibly) applies to more realistic glassy models. Interestingly enough, some features of this scenario can be understood recurring to thermometric considerations.