Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution

Using a Maximum Entropy Production Principle (MEPP), we derive a new type of relaxation equations for two-dimensional turbulent flows in the case where a prior vorticity distribution is prescribed instead of the Casimir constraints [R. Ellis, K. Haven, B. Turkington, Nonlinearity 15, 239 (2002)]. The particular case of a Gaussian prior is specifically treated in connection to minimum enstrophy states and Fofonoff flows. These relaxation equations are compared with other relaxation equations proposed by Robert and Sommeria [Phys. Rev. Lett. 69, 2776 (1992)] and Chavanis [Physica D 237, 1998 (2008)]. They can serve as numerical algorithms to compute maximum entropy states and minimum enstrophy states with appropriate constraints. We perform numerical simulations of these relaxation equations in order to illustrate geometry induced phase transitions in geophysical flows.     

Statistical mechanics of Fofonoff flows in an oceanic basin

We study the minimization of potential enstrophy at fixed circulation and energy in an oceanic basin with arbitrary topography. For illustration, we consider a rectangular basin and a linear topography h = by which represents either a real bottom topography or the β-effect appropriate to oceanic situations. Our minimum enstrophy principle is motivated by different arguments of statistical mechanics reviewed in the article. It leads to steady states of the quasigeostrophic (QG) equations characterized by a linear relationship between potential vorticity q and stream function ψ. For low values of the energy, we recover Fofonoff flows [J. Mar. Res. 13, 254 (1954)] that display a strong westward jet. For large values of the energy, we obtain geometry induced phase transitions between monopoles and dipoles similar to those found by Chavanis and Sommeria [J. Fluid Mech. 314, 267 (1996)] in the absence of topography. In the presence of topography, we recover and confirm the results obtained by Venaille and Bouchet [Phys. Rev. Lett. 102, 104501 (2009)] using a different formalism. In addition, we introduce relaxation equations towards minimum potential enstrophy states and perform numerical simulations to illustrate the phase transitions in a rectangular oceanic basin with linear topography (or β-effect).     

Statistical mechanics of two-dimensional point vortices: relaxation equations and strong mixing limit

We complement the literature on the statistical mechanics of point vortices in two-dimensional hydrodynamics. Using a maximum entropy principle, we determine the multi-species Boltzmann-Poisson equation and establish a form of Virial theorem. Using a maximum entropy production principle (MEPP), we derive a set of relaxation equations towards statistical equilibrium. These relaxation equations can be used as a numerical algorithm to compute the maximum entropy state. We mention the analogies with the Fokker-Planck equations derived by Debye and Hückel for electrolytes. We then consider the limit of strong mixing (or low energy). To leading order, the relationship between the vorticity and the stream function at equilibrium is linear and the maximization of the entropy becomes equivalent to the minimization of the enstrophy. This expansion is similar to the Debye-Hückel approximation for electrolytes, except that the temperature is negative instead of positive so that the effective interaction between like-sign vortices is attractive instead of repulsive. This leads to an organization at large scales presenting geometry-induced phase transitions, instead of Debye shielding. We compare the results obtained with point vortices to those obtained in the context of the statistical mechanics of continuous vorticity fields described by the Miller-Robert-Sommeria (MRS) theory. At linear order, we get the same results but differences appear at the next order. In particular, the MRS theory predicts a transition between sinh and tanh-like ω ? ψ relationships depending on the sign of Ku ? 3 (where Ku is the Kurtosis) while there is no such transition for point vortices which always show a sinh-like ω ? ψ relationship. We derive the form of the relaxation equations in the strong mixing limit and show that the enstrophy plays the role of a Lyapunov functional.     

Conformational temperature characterizing the folding of a protein

The time sequences of the molecular dynamics simulation for the folding process of a protein is analyzed with the inherent structure landscape which focuses on the configurational dynamics of the system. Time-dependent energy and entropy for inherent structures are introduced, and from these quantities a conformational temperature is defined. The conformational temperature follows the time evolution of a slow relaxation process and reaches the bath temperature when the system is equilibrated. We show that the nonequilibrium system is described by two temperatures, one for fast vibration and the other for slow configurational relaxation, while the equilibrium system is described by one temperature. The proposed formalism is applicable widely for systems with many metastable states.     

Deterministic equations of motion and phase ordering dynamics

We numerically solve microscopic deterministic equations of motion for the two-dimensional straight phi(4) theory with random initial states. Phase ordering dynamics is investigated. Dynamic scaling is found and it is dominated by a fixed point corresponding to the minimum energy of random initial states.     

Calculation of thermodynamic properties of substances in metastable and labile regions of real gas states

Questions related to the concepts of the equilibrium and stability of a thermodynamic system are considered. Metastable states are defined as fully equilibrium but relatively unstable states. A preferable structure of the equation of state (ES) of a single-component substance has been determined. Equations for the second virial coefficient and ES of a real gas describing the thermodynamic characteristics in a wide range of state parameters within the limits of the error in experimental data have been obtained. This was done with the help of a combined spherically symmetric potential of interaction. With the use of rigorous equations of thermodynamics, it has been found that the isochoric heat capacity in the metastable and labile regions of states remains finite and positive (with the exception of the critical point). It has been shown that, at the transition from the stable region to the region of metastable and labile states, some thermodynamic characteristics such as the isochoric heat capacity, internal energy, enthalpy, and entropy do not have any singularities. Some examples of calculation of these characteristics in the stable, metastable, and labile regions with the help of equations presented by the authors are given. They confirm the validity of the analysis performed in this paper.     

A stochastic theory of non-ideal behaviour

An approximate theory is proposed for the thermal relaxation of condensed binary [A, B] systems. During the relaxation the composition varies, or is kept constant. (The former case corresponds to an Ising lattice, the latter to a binary alloy, for example.) A variation of the composition is described with the help of a stochastic exchange A?B (as in the Glauber Model), but the microscopic transition probabilities are replaced by smoothed probabilities which depend on the system's composition and internal energy, as proposed in Part I. The variation of the internal energy is estimated with the help of a dynamic counter-part of the quasi-chemical approximation. The equations can be integrated numerically, describing the time dependence of the system's composition (if varying), internal energy, entropy and of intensive parameters corresponding to transient ‘quasi-equilibria’. The theory is applied to an Ising lattice undergoing heating or cooling through the critical temperature and compared with Monte Carlo simulation of identical processes. The agreement is judged to be very good, especially since it involves no parameter fitting. The relaxation of an [A, B] system of fixed composition, undergoing cooling to inside the co-existence curve is also studied. Three alternative lines for the evolution of internal energy are estimated: (i) unstable, with no decomposition into two phases; (ii) metastable, with spinodal decomposition and (iii) stable, with binodal decomposition. Comparison with a computer simulation of the process indicates that the initial relaxation (up to order of 100 trial exchanges per particle, depending on the system's size) fits the metastable line, the initial stage being followed by an extremely slow crossover to the stable line.     

The second law of thermodynamics under unitary evolution and external operations

The von Neumann entropy cannot represent the thermodynamic entropy of equilibrium pure states in isolated quantum systems. The diagonal entropy, which is the Shannon entropy in the energy eigenbasis at each instant of time, is a natural generalization of the von Neumann entropy and applicable to equilibrium pure states. We show that the diagonal entropy is consistent with the second law of thermodynamics upon arbitrary external unitary operations. In terms of the diagonal entropy, thermodynamic irreversibility follows from the facts that quantum trajectories under unitary evolution are restricted by the Hamiltonian dynamics and that the external operation is performed without reference to the microscopic state of the system.     

Stable and metastable states in a mesoscopic superconducting “eight” loop in presence of an external magnetic field

The stable and metastable states of different configurations of a mesoscopic loop in the form of an eight is studied in the presence of a magnetic field. We find that for certain configurations the current is equal to zero for any value of the magnetic field leading to a magnetic field independent superconducting state. The state with fixed phase circulation becomes unstable when the momentum of the superconducting electrons reaches a critical value. At this moment the kinetic energy of the superconducting condensate becomes of the same order as the potential energy of the Cooper pairs and it leads to an instability. Numerical analysis of the time-dependent Ginzburg–Landau equations shows that the absolute value of the order parameter changes gradually at the transition from a state with one phase circulation to another although the vorticity change occurs abruptly.     

Preparation and relaxation of very stable glassy states of a simulated liquid

We prepare metastable glassy states in a model glass former made of Lennard-Jones particles by sampling biased ensembles of trajectories with low dynamical activity. These trajectories form an inactive dynamical phase whose "fast" vibrational degrees of freedom are maintained at thermal equilibrium by contact with a heat bath, while the "slow" structural degrees of freedom are located in deep valleys of the energy landscape. We examine the relaxation to equilibrium and the vibrational properties of these metastable states. The glassy states we prepare by our trajectory sampling method are very stable to thermal fluctuations and also more mechanically rigid than low-temperature equilibrated configurations.     

Energy dissipation and stability of propagating surfaces

Thermodynamic equilibrium states are given by the minimum of a convex free energy function with suitable boundary conditions. Nonconvexity may lead to the coexistence of several phases and the classical Gibbs phase rule allows constructing their equilibrium properties (e.g., density or pressure). Within the framework of nonequilibrium thermodynamics, the maximization of energy dissipation (under suitable boundary conditions) can be used as an extremal principle to find stationary states. We show that stationary states generally exist for convex energy dissipation functions and that nonconvexity leads to metastable and unstable states. A geometric argument, similar in spirit to Gibbs' double-tangent construction, yields the stability limits of stationary states. This argument is applied to study a classical problem of materials science, namely the motion of a grain boundary under the influence of solute drag.     

Nonlinear mechanical response of supercooled melts under applied forces

We review recent progress on a microscopic theoretical approach to describe the nonlinear response of glass-forming colloidal dispersions under strong external forcing leading to homogeneous and inhomogeneous flow. Using mode-coupling theory (MCT), constitutive equations for the rheology of viscoelastic shear-thinning fluids are obtained. These are, in suitably simplified form, employed in continuum fluid dynamics, solved by a hybrid-Lattice Boltzmann (LB) algorithm that was developed to deal with long-lasting memory effects. The combined microscopic theoretical and mesoscopic numerical approach captures a number of phenomena far from equilibrium, including the yielding of metastable states, process-dependent mechanical properties, and inhomogeneous pressure-driven channel flow.     

Thermodynamic model for feedback control of systems with uncertain macroscopic states

Thermodynamics of feedback control processes, including the minimum work consumption of measurement, work extraction, and erasure processes of thermodynamic small systems have been investigated by researchers. We take systems with uncertain macroscopic states as the study object and study the feedback control processes of nonequilibrium macroscopic systems considering both the information entropy of microscopic states and macroscopic states. First we consider a system set that consists of systems with several macroscopic states and discuss the relations among the average information entropy of the system set, the thermodynamic entropy of the systems and the information entropy of macroscopic states of the systems. Then, we derive the expression of the average maximum net work obtained through feedback control, which relates to the free energy of the systems and the minimum work consumption of the measurement and erasure processes.     

Metastable states of spin glasses on random thin graphs

In this paper we calculate the mean number of metastable states for spin glasses on so called random thin graphs with couplings taken from a symmetric binary distribution . Thin graphs are graphs where the local connectivity of each site is fixed to some value c. As in totally connected mean field models we find that the number of metastable states increases exponentially with the system size. Furthermore we find that the average number of metastable states decreases as c in agreement with previous studies showing that finite connectivity corrections of order 1/c increase the number of metastable states with respect to the totally connected mean field limit. We also prove that the average number of metastable states in the limit is finite and converges to the average number of metastable states in the Sherrington-Kirkpatrick model. An annealed calculation for the number of metastable states of energy E is also carried out giving a lower bound on the ground state energy of these spin glasses. For small c one may obtain analytic expressions for . Received 14 October 1999 and Received in final form 14 December 1999     

Polar nano-rods under shear: From equilibrium to chaos

The orientational dynamics of rod-like particles with permanent (electric or magnetic) dipole moments in a plane Couette shear flow is investigated using mesoscopic relaxation equations combined with a generalized Landau free energy. The free energy contribution due to the coupling between average alignment and dipole orientation is derived on a microscopic basis. Numerical results of the resulting eight-dimensional dynamical system are presented for the case of longitudinal dipoles and thermodynamic conditions where the equilibrium state is a (polar or non-polar) nematic. Solution diagrams reveal presence of a large variety of periodic, transient chaotic, and chaotic dynamic states of the average alignment and dipole moment, respectively, appearing as a function of Deborah number and tumbling parameter. Compared to rods without dipoles we observe a significant preference of out-of-plane kayaking-tumbling states and, generally, a higher sensitivity to the initial conditions including bistability. We also demonstrate that the average (electric) dipole moment characterizing most of the observed states yields electrodynamic (magnetic) fields of measurable strength.     

A study of metastability in the Ising model

We consider a two-dimensional Ising ferromagnet with (+) boundary conditions and negative external field, where a Markovian time evolution is assumed. We construct, suitably restricting the allowed configurations att=0, a non equilibrium state with positive magnetization such that:

1. only one phase is present,
2. the relaxation time for unit volume is finite and can be made very large.

These results are obtained following a general method for describing metastable states proposed by Lebowitz and Penrose and exploiting the analysis of the Ising-spin-configurations in terms of contours given by Minlos and Sinai.     

Excitonic effects in core-excitation spectra of semiconductors

Computer simulations of a model glass-forming system are presented, which study the correlation between the dynamics in real space and the topography of the potential energy landscape. This analysis clearly reveals that in the supercooled regime the dynamics is strongly influenced by the presence of deep valleys in the energy landscape, corresponding to long-lived metastable amorphous states. We explicitly relate nonexponential relaxation effects and dynamic heterogeneities to these metastable states and thus to the specific topography of the energy landscape.     

Stable and metastable states and the formation and destruction of breathers in the discrete nonlinear Schrödinger equation

Statistical mechanics explains many localization phenomena of lattices such as the discrete nonlinear Schrödinger equation. However, numerical simulations show that the complete thermalization is rarely achieved. Instead, one observes metastable statistical states that are robust when excited locally. This paper investigates thermodynamically metastable states where the trajectory is confined to some part of the energy shell. The partition function and the entropy are computed with a perturbation method. This method is applicable to stable and metastable states, and it allows us to give approximative analytic expressions for the entropy in the complete thermodynamic state space.