Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics

We continue the study of a model for heat conduction [6] consisting of a chain of non-linear oscillators coupled to two Hamiltonian heat reservoirs at different temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state. Ingredients of the proof are the reduction of the infinite dimensional dynamics to a finite-dimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators. Received: 12 March 2001 / Accepted: 5 August 2001     

Fluctuation theorem in quantum heat conduction

We consider steady-state heat conduction across a quantum harmonic chain connected to reservoirs modeled by infinite collection of oscillators. The heat, Q, flowing across the oscillator in a time interval tau is a stochastic variable and we study the probability distribution function P(Q). We compute the exact generating function of Q at large tau and the large deviation function. The generating function has a symmetry satisfying the steady-state fluctuation theorem without any quantum corrections. The distribution P(Q) is non-Gaussian with clear exponential tails. The effect of finite tau and nonlinearity is considered in the classical limit through Langevin simulations. We also obtain the prediction of quantum heat current fluctuations at low temperatures in clean wires.     

Ecological optimization of an irreversible harmonic oscillators Carnot heat engine

A model of an irreversible quantum Carnot heat engine with heat resistance, internal irreversibility and heat leakage and many non-interacting harmonic oscillators is established in this paper. Based on the quantum master equation and semi-group approach, equations of some important performance parameters, such as power output, efficiency, exergy loss rate and ecological function for the irreversible quantum Carnot heat engine are derived. The optimal ecological performance of the heat engine in the classical limit is analyzed with numerical examples. Effects of internal irreversibility and heat leakage on the ecological performance are discussed. A performance comparison of the quantum heat engine under maximum ecological function and maximum power conditions is also performed.     

Equilibrium and stationary nonequilibrium states in a chain of colliding harmonic oscillators

Equilibrium and nonequilibrium properties of a chain of colliding harmonic oscillators (ding-dong model) are investigated. Our chain is modeled as harmonically bounded particles that can only interact with neighboring particles by hard-core interaction. Between the collisions, particles are just independent harmonic oscillators. We are especially interested in the stationary nonequilibrium state of the ding-dong model coupled with two stochastic heat reservoirs (not thermostated) at the ends, whose temperature is different. We check the Gallavotti-Cohen fluctuation theorem [G. Gallavoti and E. G. D. Cohen, Phys. Rev. Lett. 74, 2694 (1995)] and also the Evans-Searles identity [D. Evans and D. Searles, Phys. Rev. E. 50, 1994 (1994)] numerically. It is verified that the former theorem is satisfied for this system, although the system is not a thermostated system.     

Statistical Physics on the Space (x, v) for Dissipative Systems and Study of an Ensemble of Harmonic Oscillators in a Weak Linear Dissipative Medium

We use the phase space position-velocity (x, v) to deal with the statistical properties of velocity dependent dynamical systems, like dissipative ones. Within this approach, we study the statistical properties of an ensemble of harmonic oscillators in a linear weak dissipative media. Using the Debye model of a crystal, we calculate at first order in the dissipative parameter the entropy, free energy, internal energy, equation of state and specific heat using the classical and quantum approaches. For the classical approach we found that the entropy, the equation of state, and the free energy depend on the dissipative parameter, but the internal energy and specific heat do not depend of it. For the quantum case, we found that all the thermodynamical quantities depend on this parameter. PACS: 05.20.Gg, 05.30.Ch, 05.20.-y, 05.30.-d     

Effect of noise and perturbations on limit cycle systems

This paper demonstrates that the influence of noise and of external perturbations on a nonlinear oscillator can vary strongly along the limit cycle and upon transition from limit cycle to stationary point behaviour. For this purpose we consider the role of noise on the Bonhoeffer-van der Pol model in a range of control parameters where the model exhibits a limit cycle, but the parameters are close to values corresponding to a stable stationary point. Our analysis is based on the van Kampen approximation for solutions of the Fokker-Planck equation in the limit of weak noise. We investigate first separately the effect of noise on motion tangential and normal to the limit cycle. The key result is that noise induces diffusion-like spread along the limit cycle, but quasistationary behaviour normal to the limit cycle. We then describe the coupled motion and show that noise acting in the normal direction can strongly enhance diffusion along the limit cycle. We finally argue that the variability of the system's response to noise can be exploited in populations of nonlinear oscillators in that weak coupling can induce synchronization as long as the single oscillators operate in a regime close to the transition between oscillatory and excitatory modes.     

Heat Transport in Harmonic Lattices

We work out the non-equilibrium steady state properties of a harmonic lattice which is connected to heat reservoirs at different temperatures. The heat reservoirs are themselves modeled as harmonic systems. Our approach is to write quantum Langevin equations for the system and solve these to obtain steady state properties such as currents and other second moments involving the position and momentum operators. The resulting expressions will be seen to be similar in form to results obtained for electronic transport using the non-equilibrium Green’s function formalism. As an application of the formalism we discuss heat conduction in a harmonic chain connected to self-consistent reservoirs. We obtain a temperature dependent thermal conductivity which, in the high-temperature classical limit, reproduces the exact result on this model obtained recently by Bonetto, Lebowitz and Lukkarinen.     

Dynamical response of nanostructures and Joule heat release

We consider Joule heat release in a quantum wire joining two classical reservoirs under the action of a nonstationary periodic electric field. The rate of heat generation and its spatial distribution is discussed. The heat is spread over the lengths of electron mean free paths in the reservoirs. We find that the total rates of heat generation in both reservoirs that are joined by the nanostructure are the same.     

A perturbative nonequilibrium renormalization group method for dissipative quantum mechanics

We study a generic problem of dissipative quantum mechanics, a small local quantum system with discrete states coupled in an arbitrary way (i.e. not necessarily linear) to several infinitely large particle or heat reservoirs. For both bosonic or fermionic reservoirs we develop a quantum field-theoretical diagrammatic formulation in Liouville space by expanding systematically in the reservoir-system coupling and integrating out the reservoir degrees of freedom. As a result we obtain a kinetic equation for the reduced density matrix of the quantum system. Based on this formalism, we present a formally exact perturbative renormalization group (RG) method from which the kernel of this kinetic equation can be calculated. It is demonstrated how the nonequilibrium stationary state (induced by several reservoirs kept at different chemical potentials or temperatures), arbitrary observables such as the transport current, and the time evolution into the stationary state can be calculated. Most importantly, we show how RG equations for the relaxation and dephasing rates can be derived and how they cut off generically the RG flow of the vertices. The method is based on a previously derived real-time RG technique [1-4] but formulated here in Laplace space and generalized to arbitrary reservoir-system couplings. Furthermore, for fermionic reservoirs with flat density of states, we make use of a recently introduced cutoff scheme on the imaginary frequency axis [5] which has several technical advantages. Besides the formal set-up of the RG equations for generic problems of dissipative quantum mechanics, we demonstrate the method by applying it to the nonequilibrium isotropic Kondo model. We present a systematic way to solve the RG equations analytically in the weak-coupling limit and provide an outlook of the applicability to the strong-coupling case.     

Matrix-product states for a one-dimensional lattice gas with parallel dynamics

The hopping motion of classical particles on a chain coupled to reservoirs at both ends is studied for parallel dynamics with arbitrary probabilities. The stationary state is obtained in the form of an alternating matrix product. The properties of one- and two-dimensional representations are studied in detail and a general relation of the matrix algebra to that of the sequential limit is found. In this way the general phase diagram of the model is obtained. The mechanism of the sequential limit, the formulation as a vertex model, and other aspects are discussed.     

Nonequilibrium invariant measure under heat flow

We provide an explicit representation of the nonequilibrium invariant measure for a chain of harmonic oscillators with conservative noise in the presence of stationary heat flow. By first determining the covariance matrix, we are able to express the measure as the product of Gaussian distributions aligned along some collective modes that are spatially localized with power-law tails. Numerical studies show that such a representation applies also to a purely deterministic model, the quartic Fermi-Pasta-Ulam chain.     

Fourier’s Law for a Microscopic Model of Heat Conduction

We consider a chain of N harmonic oscillators perturbed by a conservative stochastic dynamics and coupled at the boundaries to two gaussian thermostats at different temperatures. The stochastic perturbation is given by a diffusion process that exchange momentum between nearest neighbor oscillators conserving the total kinetic energy. The resulting total dynamics is a degenerate hypoelliptic diffusion with a smooth stationary state. We prove that the stationary state, in the limit as N→ ∞, satisfies Fourier’s law and the linear profile for the energy average     

A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities

We introduce a multidimensional peridynamic formulation for transient heat-transfer. The model does not contain spatial derivatives and uses instead an integral over a region around a material point. By construction, the formulation converges to the classical heat transfer equations in the limit of the horizon (the nonlocal region around a point) going to zero. The new model, however, is suitable for modeling, for example, heat flow in bodies with evolving discontinuities such as growing insulated cracks. We introduce the peridynamic heat flux which exists even at sharp corners or when the isotherms are not smooth surfaces. The peridynamic heat flux coincides with the classical one in simple cases and, in general, it converges to it in the limit of the peridynamic horizon going to zero. We solve test problems and compare results with analytical solutions of the classical model or with other numerical solutions. Convergence to the classical solutions is seen in the limit of the horizon going to zero. We then solve the problem of transient heat flow in a plate in which insulated cracks grow and intersect thus changing the heat flow patterns. We also model heat transfer in a fiber-reinforced composite and observe transient but steep thermal gradients at the interfaces between the highly conductive fibers and the low conductivity matrix. Such thermal gradients can lead to delamination cracks in composites from thermal fatigue. The formulation may be used to, for example, evaluate effective thermal conductivities in bodies with an evolving distribution of insulating or permeable, possibly intersecting, cracks of arbitrary shapes.     

Non-equilibrium effects upon the non-Markovian Caldeira–Leggett quantum master equation

We obtain a non-Markovian quantum master equation directly from the quantization of a non-Markovian Fokker–Planck equation describing the Brownian motion of a particle immersed in a generic environment (e.g. a non-thermal fluid). As far as the especial case of a heat bath comprising of quantum harmonic oscillators is concerned, we derive a non-Markovian Caldeira–Leggett master equation on the basis of which we work out the concept of non-equilibrium quantum thermal force exerted by the harmonic heat bath upon the Brownian motion of a free particle. The classical limit (or dequantization process) of this sort of non-equilibrium quantum effect is scrutinized, as well.     

Decoherence in strongly coupled quantum oscillators

In this paper, we present a comprehensive analysis of the coherence phenomenon of two coupled dissipative oscillators. The action of a classical driving field on one of the oscillators is also analyzed. Master equations are derived for both regimes of weakly and strongly interacting oscillators from which interesting results arise concerning the coherence properties of the joint and the reduced system states. The strong coupling regime is required to achieve a large frequency shift of the oscillator normal modes, making it possible to explore the whole profile of the spectral density of the reservoirs. We show how the decoherence process may be controlled by shifting the normal mode frequencies to regions of small spectral density of the reservoirs. Different spectral densities of the reservoirs are considered and their effects on the decoherence process are analyzed. For oscillators with different damping rates, we show that the worse-quality system is improved and vice versa, a result which could be useful for quantum state protection. State recurrence and swap dynamics are analyzed as well as their roles in delaying the decoherence process.     

A semi-microscopic model of the heat conduction law

We give two very simple quantum models for the heat conduction law using a master equation approach for the probability distribution of the quantum numbers of the oscillators. The probability of interaction of the oscillators is given by the Landau-Teller formula.     

Fourier's law from closure equations

We sketch a rigorous derivation of Fourier's law from a system of closure equations for a nonequilibrium stationary state of a Hamiltonian system of coupled oscillators subjected to heat baths on the boundary. The local heat flux is proportional to the temperature gradient with a temperature dependent heat conductivity and the stationary temperature exhibits a nonlinear profile.     

Modeling of heat explosion with convection

The work is devoted to numerical simulations of the interaction of heat explosion with natural convection. The model consists of the heat equation with a nonlinear source term describing heat production due to an exothermic chemical reaction coupled with the Navier-Stokes equations under the Boussinesq approximation. We show how complex regimes appear through successive bifurcations leading from a stable stationary temperature distribution without convection to a stationary symmetric convective solution, stationary asymmetric convection, periodic in time oscillations, and finally aperiodic oscillations. A simplified model problem is suggested. It describes the main features of solutions of the complete problem.     

Nontrivial properties of heat flow: Analytical study of some anharmonic lattice microscopic models

We address the analytical investigation of nontrivial properties of heat flow, starting from microscopic models of matter. We present an integral representation for the expression of the heat flow, by taking as our crystal model a self-consistent chain of anharmonic oscillators, precisely, a chain of oscillators with harmonic interparticle interactions, anharmonic on-site potentials, thermal reservoirs at the boundaries, and still with some residual inner stochastic baths. We propose an approximative scheme to make the integral formalism analytically treatable: we test the approximations in harmonic models and analyze some anharmonic systems. For the case of graded anharmonic models with weak interparticle interactions, we derive an expression for the thermal conductivity, and show the existence of thermal rectification and also nonnegative differential thermal resistance. The details of our formalism allow us to note the ingredients behind these phenomena, and the generality of our results (i.e., the results will be valid for other interactions and regimes) shows that rectification in graded materials is a ubiquitous property.     

Entropy Production in Nonlinear,Thermally Driven Hamiltonian Systems

We consider a finite chain of nonlinear oscillators coupled at its ends to two infinite heat baths which are at different temperatures. Using our earlier results about the existence of a stationary state, we show rigorously that for arbitrary temperature differences and arbitrary couplings, such a system has a unique stationary state. (This extends our earlier results for small temperature differences.) In all these cases, any initial state will converge (at an unknown rate) to the stationary state. We show that this stationary state continually produces entropy. The rate of entropy production is strictly negative when the temperatures are unequal and is proportional to the mean energy flux through the system