Orbital measures in non-equilibrium statistical mechanics: the Onsager relations

We assume that the properties of nonequilibrium stationary states of systems of particles can be expressed in terms of weighted orbital measures, i.e. through periodic orbit expansions. This allows us to derive the Onsager relations for systems of $N$ particles subject to a Gaussian thermostat, under the assumption that the entropy production rate is equal to the phase space contraction rate. Moreover, this also allows us to prove that the relevant transport coefficients are not negative. In the appendix we give an argument for the proper way of treating grazing collisions, a source of possible singularities in the dynamics.Comment: LaTeX, 14 pages, 1 TeX figure in the tex

The Gallavotti-Cohen Fluctuation Theorem for a non-chaotic model

We test the applicability of the Gallavotti-Cohen fluctuation formula on a nonequilibrium version of the periodic Ehrenfest wind-tree model. This is a one-particle system whose dynamics is rather complex (e.g. it appears to be diffusive at equilibrium), but its Lyapunov exponents are nonpositive. For small applied field, the system exhibits a very long transient, during which the dynamics is roughly chaotic, followed by asymptotic collapse on a periodic orbit. During the transient, the dynamics is diffusive, and the fluctuations of the current are found to be in agreement with the fluctuation formula, despite the lack of real hyperbolicity. These results also constitute an example which manifests the difference between the fluctuation formula and the Evans-Searles identity.Comment: 12 pages, submitted to Journal of Statistical Physic

Gibbs entropy and irreversible thermodynamics

Recently a number of approaches has been developed to connect the microscopic dynamics of particle systems to the macroscopic properties of systems in nonequilibrium stationary states, via the theory of dynamical systems. This way a direct connection between dynamics and Irreversible Thermodynamics has been claimed to have been found. However, the main quantity used in these studies is a (coarse-grained) Gibbs entropy, which to us does not seem suitable, in its present form, to characterize nonequilibrium states. Various simplified models have also been devised to give explicit examples of how the coarse-grained approach may succeed in giving a full description of the Irreversible Thermodynamics. We analyze some of these models pointing out a number of difficulties which, in our opinion, need to be overcome in order to establish a physically relevant connection between these models and Irreversible Thermodynamics.Comment: 19 pages, 4 eps figures, LaTeX2

Entropy production and coarse-graining in Markov processes

We study the large time fluctuations of entropy production in Markov processes. In particular, we consider the effect of a coarse-graining procedure which decimates {\em fast states} with respect to a given time threshold. Our results provide strong evidence that entropy production is not directly affected by this decimation, provided that it does not entirely remove loops carrying a net probability current. After the study of some examples of random walks on simple graphs, we apply our analysis to a network model for the kinesin cycle, which is an important biomolecular motor. A tentative general theory of these facts, based on Schnakenberg's network theory, is proposed.Comment: 18 pages, 13 figures, submitted for publicatio

Harmonic damped oscillators with feedback. A Langevin study

We consider a system in direct contact with a thermal reservoir and which, if left unperturbed, is well described by a memory-less equilibrium Langevin equation of the second order in the time coordinate. In such conditions, the strength of the noise fluctuations is set by the damping factor, in accordance with the Fluctuation and Dissipation theorem. We study the system when it is subject to a feedback mechanism, by modifying the Langevin equation accordingly. Memory terms now arise in the time evolution, which we study in a non-equilibrium steady state. Two types of feedback schemes are considered, one focusing on time shifts and one on phase shifts, and for both cases we evaluate the power spectrum of the system's fluctuations. Our analysis finds application in feedback cooled oscillators, such as the Gravitational Wave detector AURIGA.Comment: 17 page

Entropy production and coarse-graining in Markov processes

We study the large time fluctuations of entropy production in Markov processes. In particular, we consider the effect of a coarse-graining procedure which decimates {\em fast states} with respect to a given time threshold. Our results provide strong evidence that entropy production is not directly affected by this decimation, provided that it does not entirely remove loops carrying a net probability current. After the study of some examples of random walks on simple graphs, we apply our analysis to a network model for the kinesin cycle, which is an important biomolecular motor. A tentative general theory of these facts, based on Schnakenberg's network theory, is proposed.Comment: 18 pages, 13 figures, submitted for publicatio

Note on Phase Space Contraction and Entropy Production in Thermostatted Hamiltonian Systems

The phase space contraction and the entropy production rates of Hamiltonian systems in an external field, thermostatted to obtain a stationary state are considered. While for stationary states with a constant kinetic energy the two rates are formally equal for all numbers of particles N, for stationary states with constant total (kinetic and potential) energy this only obtains for large N. However, in both cases a large number of particles is required to obtain equality with the entropy production rate of Irreversible Thermodynamics. Consequences of this for the positivity of the transport coefficients and for the Onsager relations are discussed. Numerical results are presented for the special case of the Lorentz gas.Comment: 16 pages including 1 table and 3 figures. LaTeX forma

Work and Thermal Fluctuations in Crystal Indentation under Deterministic and Stochastic Thermostats: The Role of System–Bath Coupling

The Jarzynski equality (JE) was originally derived under the deterministic Hamiltonian formalism, and later, it was demonstrated that stochastic Langevin dynamics also lead to the JE. However, the JE has been verified mainly in small, low-dimensional systems described by Langevin dynamics. Although the two theoretical derivations apparently lead to the same expression, we illustrate that they describe fundamentally different experimental conditions. While the Hamiltonian framework assumes that the thermal bath producing the initial canonical equilibrium switches off for the duration of the work process, the Langevin bath effectively acts on the system. Moreover, the former considers an environment with which the system may interact, whereas the latter does not. In this study, we investigate the effect of the bath on the measurable quantity of the JE through molecular dynamics simulations of crystal nanoindentation employing deterministic and stochastic thermostats. Our analysis shows that the distributions of the kinetic energy and the mechanical work produced during the indentation processes are affected by the interaction between the system and the thermostat baths. As a result, the type of thermostatting has also a clear effect on the left-hand side of the JE, which enables the estimation of the free-energy difference characterizing the process

Spatiotemporal chaos and the dynamics of coupled Langmuir and ion-acoustic waves in plasmas

A simulation study is performed to investigate the dynamics of coupled Langmuir waves (LWs) and ion-acoustic waves (IAWs) in an unmagnetized plasma. The effects of dispersion due to charge separation and the density nonlinearity associated with the IAWs, are considered to modify the properties of Langmuir solitons, as well as to model the dynamics of relatively large amplitude wave envelopes. It is found that the Langmuir wave electric field, indeed, increases by the effect of ion-wave nonlinearity (IWN). Use of a low-dimensional model, based on three Fourier modes shows that a transition to temporal chaos is possible, when the length scale of the linearly excited modes is larger than that of the most unstable ones. The chaotic behaviors of the unstable modes are identified by the analysis of Lyapunov exponent spectra. The space-time evolution of the coupled LWs and IAWs shows that the IWN can cause the excitation of many unstable harmonic modes, and can lead to strong IAW emission. This occurs when the initial wave field is relatively large or the length scale of IAWs is larger than the soliton characteristic size. Numerical simulation also reveals that many solitary patterns can be excited and generated through the modulational instability (MI) of unstable harmonic modes. As time goes on, these solitons are seen to appear in the spatially partial coherence (SPC) state due to the free ion-acoustic radiation as well as in the state of spatiotemporal chaos (STC) due to collision and fusion in the stochastic motion. The latter results the redistribution of initial wave energy into a few modes with small length scales, which may lead to the onset of Langmuir turbulence in laboratory as well as space plasmas.Comment: 10 Pages, 14 Figures; to appear in Physical Review