Causality, response theory, and the second law of thermodynamics

We show that there is a close connection between the assumption of causality and the second law of thermodynamics. We also show that for a class of classical reversible deterministic systems it is overwhelmingly improbable either to find causal steady states that violate the second law, or anticausal states that satisfy the second law. These arguments indicate that the existence of (and the sign associated with) the second law of thermodynamics is ultimately determined by causality. Our discussion employs a Green-Kubo relation that we derive for an anticausal linear transport coefficient

Fluctuations in Nonequilibrium Statistical Mechanics: Models, Mathematical Theory, Physical Mechanisms

The fluctuations in nonequilibrium systems are under intense theoretical and experimental investigation. Topical ``fluctuation relations'' describe symmetries of the statistical properties of certain observables, in a variety of models and phenomena. They have been derived in deterministic and, later, in stochastic frameworks. Other results first obtained for stochastic processes, and later considered in deterministic dynamics, describe the temporal evolution of fluctuations. The field has grown beyond expectation: research works and different perspectives are proposed at an ever faster pace. Indeed, understanding fluctuations is important for the emerging theory of nonequilibrium phenomena, as well as for applications, such as those of nanotechnological and biophysical interest. However, the links among the different approaches and the limitations of these approaches are not fully understood. We focus on these issues, providing: a) analysis of the theoretical models; b) discussion of the rigorous mathematical results; c) identification of the physical mechanisms underlying the validity of the theoretical predictions, for a wide range of phenomena.Comment: 44 pages, 2 figures. To appear in Nonlinearity (2007

Master equation approach to the conjugate pairing rule of Lyapunov spectra for many-particle thermostatted systems

The master equation approach to Lyapunov spectra for many-particle systems is applied to non-equilibrium thermostatted systems to discuss the conjugate pairing rule. We consider iso-kinetic thermostatted systems with a shear flow sustained by an external restriction, in which particle interactions are expressed as a Gaussian white randomness. Positive Lyapunov exponents are calculated by using the Fokker-Planck equation to describe the tangent vector dynamics. We introduce another Fokker-Planck equation to describe the time-reversed tangent vector dynamics, which allows us to calculate the negative Lyapunov exponents. Using the Lyapunov exponents provided by these two Fokker-Planck equations we show the conjugate pairing rule is satisfied for thermostatted systems with a shear flow in the thermodynamic limit. We also give an explicit form to connect the Lyapunov exponents with the time-correlation of the interaction matrix in a thermostatted system with a color field.Comment: 10 page

Pod systems : an equivariant ordinary differential equation approach to dynamical systems on a spatial domain

We present a class of systems of ordinary differential equations (ODEs), which we call 'pod systems', that offers a new perspective on dynamical systems defined on a spatial domain. Such systems are typically studied as partial differential equations, but pod systems bring the analytic techniques of ODE theory to bear on the problems, and are thus able to study behaviours and bifurcations that are not easily accessible to the standard methods. In particular, pod systems are specifically designed to study spatial dynamical systems that exhibit multi-modal solutions. A pod system is essentially a linear combination of parametrized functions in which the coefficients and parameters are variables whose dynamics are specified by a system of ODEs. That is, pod systems are concerned with the dynamics of functions of the form psi(s, t) = y(1)(t)phi(s; x(1)(t)) + ... + y(N)(t)phi(s; x(N)(t)), where s is an element of R-n is the spatial variable and phi: R-n x R-d -> R. The parameters x(i) is an element of R-d and coefficients y(i) is an element of R are dynamic variables which evolve according to some system of ODEs,. x(i) = G(i)(x, y) and y(i) = H-i(x, y), for i = 1,..., N. The dynamics of psi in function space can then be studied through the dynamics of the x and y in finite dimensions. A vital feature of pod systems is that the ODEs that specify the dynamics of the x and y variables are not arbitrary; restrictions on G(i) and H-i are required to guarantee that the dynamics of psi in function space are well defined (that is, that trajectories are unique). One important restriction is symmetry in the ODEs which arises because psi is invariant under permutations of the indices of the (x(i), y(i)) pairs. However, this is not the whole story, and the primary goal of this paper is to determine the necessary structure of the ODEs explicitly to guarantee that the dynamics of psi are well defined