Kubo formulas for relativistic fluids in strong magnetic fields

Magnetohydrodynamics of strongly magnetized relativistic fluids is derived in the ideal and dissipative cases, taking into account the breaking of spatial symmetries by a quantizing magnetic field. A complete set of transport coefficients, consistent with the Curie and Onsager principles, is derived for thermal conduction, as well as shear and bulk viscosities. It is shown that in the most general case the dissipative function contains five shear viscosities, two bulk viscosities, and three thermal conductivity coefficients. We use Zubarev’s non-equilibrium statistical operator method to relate these transport coefficients to correlation functions of the equilibrium theory. The desired relations emerge at linear order in the expansion of the non-equilibrium statistical operator with respect to the gradients of relevant statistical parameters (temperature, chemical potential, and velocity.) The transport coefficients are cast in a form that can be conveniently computed using equilibrium (imaginary-time) infrared Green’s functions defined with respect to the equilibrium statistical operator.     

What Does Dynamical Systems Theory Teach Us about Fluids?

We use molecular dynamics simulations to compute the Lyapunov spectra of many-particle systems resembling simple fluids in thermal equilibrium and in non-equilibrium stationary states. Here we review some of the most interesting results and point to open questions.     

Dynamically regulated energy barriers with violation of symmetry for reaction path

The motion of a particle on a flexible one-dimensional base is investigated numerically. It is regarded as a barrier-overcoming process in a multi-dimensional phase space. Driving the system far from equilibrium, a directional flow of the particle is observed. In order to understand the appearance of the directionality, two kinds of transition states for the barrier-overcoming process are considered particularly: one of which is for the overcoming process from left to right and the other from right to left. System's conformations for these two transitions are found to be identical at equilibrium, but become different at non-equilibrium, which means that the transition states become direction dependent. The energy barrier heights defined for the two transition states are found to account for biased reaction rates for the directional flow. Thus, the barrier heights work as activation energy extended to far from equilibrium. From the viewpoint of phase space, the directional dependence of the transition states is understood as the symmetry violation of the reaction paths, the route of the reaction being selected depending on the overcoming direction. It should be noted that such a selection is brought about by autonomous dynamics of the system itself. These results suggest the possibility to define an effective potential for a class of dynamical systems in non-equilibrium situations.     

Distinctive features of random local motions in critical and supercritical fluids

Equations for large-scale local fluctuations in fluids, from an ideal gas to an incompressible fluid, including the critical and supercritical state are derived for the first time based on the first principles. The modern phenomenological representation of the critical state of fluids is confirmed and essentially refined; in particular, it is demonstrated that that local density fluctuations in a compressible fluid are accompanied by nonthermodynamic fluctuations in the collective velocity and temperature of the fluid. Distinctive features of the development of these fluctuations near the critical point determine the specific behavior of fluids in the critical and supercritical states.     

Diffusion within a near-critical nanopore fluid

Results are presented for grand canonical Monte Carlo (GCMC) and both equilibrium and non-equilibrium molecular dynamics simulations (EMD and NEMD) conducted over a range of densities and temperatures that span the two-phase coexistence and supercritical regions for a pure fluid adsorbed within a model crystalline nanopore. The GCMC simulations provided the low temperature coexistence points for the open pore fluid and were used to locate the capillary critical temperature for the system. The equilibrium configurational states obtained from these simulations were then used as input data for the EMD simulations in which the self-diffusion coefficients were computed using the Einstein equation. NEMD colour diffusion simulations were also conducted to validate the use of a system averaged Einstein analysis for this inhomogeneous fluid. In all cases excellent agreement was observed between the equilibrium (linear response theory) predictions for the diffusivities and non-equilibrium colour diffusivities. The simulation results are also compared with a recently published quasi-hydrodynamic theory of Pozhar and Gubbins (Pozhar, L. A., and Gubbins, K. E., 1993, J. Chem. Phys., 99, 8970; 1997, Phys. Rev. E, 56, 5367.). The model fluid and the nature of the fluid wall interactions employed conform to the decomposition of the particle–particle interaction potential explicitly used by Pozhar and Gubbins. The local self-diffusivity was calculated from the local fluid–fluid and fluid wall hard core collision frequencies. While this theory provides reasonable results at moderate pore fluid densities, poor agreement is observed in the low density limit.     

Non-equilibrium thermodynamics and dissipative fluid theories

Summary Some of the available, phenomenological studies on the dissipative fluid theories have involved extending the set of independent dynamical variables. In the favourite case of a chemically inert fluid, one can propose to enlarge the usual hydrodynamic space both by introducing eight components of the stress deviator and the heat flux and by treating them as the fundamental variables on the same footing as the mass density and the specific internal energy. A candidate theory of this kind is based upon the quasi-linear, first-order partial differential equations for the evaluation of all variables. In this paper, the differential field equations are studied with a view to a deeper understanding of non-equilibrium thermodynamics for dissipative fluids. A characteristic feature of the endeavour is that not only it is now possible to have the differential field equations consistent with a supplementary balance law, interpreted as the equation of balance of entropy, but also possible to clarify the meaning of temperature and pressure beyond local equilibrium and to obtain the theory of thermodynamic potentials for systems ?not infinitesimally near to equilibrium?. These results are achieved via the use of the critical-point theory, as formulated by Morse, in the context of the well-known extremum property of entropy. Mathematically, the supplementary balance law is derived by exploiting the calculus of ?vertical? differential forms, and the differential field equations are defined intrinsically,i.e. without making any explicit reference to a particular coordinate system. Finally, the paper discusses some problems concerning the structure of an expression for the entropy flux.     

Cell-level canonical sampling by velocity scaling for multiparticle collision dynamics simulations

A local Maxwellian thermostat for the multiparticle collision dynamics algorithm is proposed. The algorithm is based on a scaling of the relative velocities of the fluid particles within a collision cell. The scaling factor is determined from the distribution of the kinetic energy within such a cell. Thereby the algorithm ensures that the distribution of the relative velocities is given by the Maxwell–Boltzmann distribution. The algorithm is particularly useful for non-equilibrium systems, where temperature has to be controlled locally. We perform various non-equilibrium simulations for fluids in shear and pressure-driven flow, which confirm the validity of the proposed simulation scheme. In addition, we determine the dynamic structure factors for fluids with and without thermostat, which exhibit significant differences due to suppression of the diffusive part of the energy transport of the isothermal system.     

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 Properties of Non-equilibrium States in Quantum Field Theory

Within the framework of relativistic quantum field theory, a novel method is established which allows for distinguishing non-equilibrium states admitting locally a thermodynamic interpretation. The basic idea is to compare these states with global equilibrium states (KMS states) by means of local thermal observables. With the help of such observables, the states can be ordered into classes of increasing local thermal stability. Moreover, it is possible to identify states exhibiting certain specific thermal properties of interest, such as a definite local temperature or entropy density. The method is illustrated in a simple model describing the spatio-temporal evolution of a “big heat bang.”     

Poiseuille flow of a micropolar fluid

We use non-equilibrium molecular dynamics simulations to study the flow of a micropolar fluid and to test an extended Navier-Stokes theory (ENS) for such fluids. The angular streaming velocity (which is of course missing in the classical Navier-Stokes theory) and the translational streaming velocity are found to be in good agreement with the predictions of ENS theory. Besides, owing to molecular rotation, the translational streaming velocity profile is shown to deviate from the classical parabolic profile. Finally, temperature profiles calculated using three different expressions (a kinetic translational, a kinetic rotational and a recently derived configurational expression) are found to be in excellent agreement, demonstrating that the equipartition principle still holds in this non-equilibrium system. No deviation from the classical quartic temperature profile is observed.     

Derivation of Maximum Entropy Principles in Two-Dimensional Turbulence via Large Deviations

The continuum limit of lattice models arising in two-dimensional turbulence is analyzed by means of the theory of large deviations. In particular, the Miller–Robert continuum model of equilibrium states in an ideal fluid and a modification of that model due to Turkington are examined in a unified framework, and the maximum entropy principles that govern these models are rigorously derived by a new method. In this method, a doubly indexed, measure-valued random process is introduced to represent the coarse-grained vorticity field. The natural large deviation principle for this process is established and is then used to derive the equilibrium conditions satisfied by the most probable macrostates in the continuum models. The physical implications of these results are discussed, and some modeling issues of importance to the theory of long-lived, large-scale coherent vortices in turbulent flows are clarified.     

Benford’s law in non-equilibrium processes: Droplet collisions case

Benford’s law is investigated for the simulation results generated from non-equilibrium molecular dynamics. A statistic to measure how closely a set of the numbers follows Benford’s law is defined. The simulation data are from the collisions of two nano droplets with different impact velocities. When a non-equilibrium system returns to its equilibrium state, some physical quantities relevant to the non-equilibrium settings follow Benford’s law more closely. The initial settings for the non-equilibrium state can be interpreted as a data fabrication of its corresponding equilibrium state. A connection with the Shannon entropy for the first digit distribution is also discussed.     

Criticality in a non-equilibrium,driven system: Charged colloidal rods (fd-viruses) in electric fields

Experiments on suspensions of charged colloidal rods (fd-virus particles) in external electric fields are performed, which show that a non-equilibrium critical point can be identified. Several transition lines of field-induced phases and states meet at this point and it is shown that there is a length- and time-scale which diverge at the non-equilibrium critical point. The off-critical and critical behavior is characterized, with both power law and logarithmic divergencies. These experiments show that analogous features of the classical, critical divergence of correlation lengths and relaxation times in equilibrium systems are also exhibited by driven systems that are far out of equilibrium, related to phases/states that do not exist in the absence of the external field.     

Natural Energy Bounds in Quantum Thermodynamics

Given a stationary state for a noncommutative flow, we study a boundedness condition, depending on a parameter β>0, which is weaker than the KMS equilibrium condition at inverse temperature β. This condition is equivalent to a holomorphic property closely related to the one recently considered by Ruelle and D'Antoni–Zsido and shared by a natural class of non-equilibrium steady states. Our holomorphic property is stronger than Ruelle's one and thus selects a restricted class of non-equilibrium steady states. We also introduce the complete boundedness condition and show this notion to be equivalent to the Pusz–Woronowicz complete passivity property, hence to the KMS condition. In Quantum Field Theory, the β-boundedness condition can be interpreted as the property that localized state vectors have energy density levels increasing β-subexponentially, a property which is similar in the form and weaker in the spirit than the modular compactness-nuclearity condition. In particular, for a Poincaré covariant net of C^*-algebras on Minkowski spacetime, the β-boundedness property,β≥ 2π, for the boosts is shown to be equivalent to the Bisognano–Wichmann property. The Hawking temperature is thus minimal for a thermodynamical system in the background of a Rindler black hole within the class of β-holomorphic states. More generally, concerning the Killing evolution associated with a class of stationary quantum black holes, we characterize KMS thermal equilibrium states at Hawking temperature in terms of the boundedness property and the existence of a translation symmetry on the horizon. Received: 2 October 2000 / Accepted: 5 December 2000     

Dynamical diversity and metastability in a hindered granular column near jamming

Granular media jam into a panoply of metastable states. The way in which these states are achieved depends on the nature of local and global constraints on grains; here we investigate this issue by means of a non-equilibrium stochastic model of a hindered granular column near its jamming limit. Grains feel the constraints of grains above and below them differently, depending on their position. A rich phase diagram with four dynamical phases (ballistic, activated, logarithmic and glassy) is revealed. The statistics of the jamming time and of the metastable states reached as attractors of the zero-temperature dynamics is investigated in each of these phases. Of particular interest is the glassy phase, where intermittency and a strong deviation from Edwards' flatness are manifest.     

Response Theory for Equilibrium and Non-Equilibrium Statistical Mechanics: Causality and Generalized Kramers-Kronig Relations

We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions entails the possibility of writing a set of Kramers-Kronig (K-K) relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey K-K relations. Specific results are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive K-K analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short-time response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey. The theory exposed in the present paper is dual to the time-dependent theory of perturbations to equilibrium states and to non-equilibrium steady states, and has in principle similar range of applicability and limitations. In order to connect the equilibrium and the non equilibrium steady state case, we show how to rewrite the classical response theory by Kubo so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. These results, taking into account the chaotic hypothesis by Gallavotti and Cohen, might be relevant in several fields, including climate research. In particular, whereas the fluctuation-dissipation theorem does not work for non-equilibrium systems, because of the non-equivalence between internal and external fluctuations, K-K relations might be robust tools for the definition of a self-consistent theory of climate change.     

Non—equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method

In this paper, we propose a new approach to implementing boundary conditions in the lattice Boltzmann method (LBM). The basic idea is to decompose the distribution function at the boundary node into its equilibrium and non-equilibrium parts, and then to approximate the non-equilibrium part with a first-order extrapolation of the non-equilibrium part of the distribution at the neighbouring fluid node. Schemes for velocity and pressure boundary conditions are constructed based on this method. The resulting schemes are of second-order accuracy. Numerical tests show that the numerical solutions of the LBM together with the present boundary schemes are in excellent agreement with the analytical solutions. Second-order convergence is also verified from the results. It is also found that the numerical stability of the present schemes is much better than that of the original extrapolation schemes proposed by Chen et al. (1996 Phys. Fluids 8 2527).     

Dissipation-Driven Selection under Finite Diffusion: Hints from Equilibrium and Separation of Time Scales

When exposed to a thermal gradient, reaction networks can convert thermal energy into the chemical selection of states that would be unfavourable at equilibrium. The kinetics of reaction paths, and thus how fast they dissipate available energy, might be dominant in dictating the stationary populations of all chemical states out of equilibrium. This phenomenology has been theoretically explored mainly in the infinite diffusion limit. Here, we show that the regime in which the diffusion rate is finite, and also slower than some chemical reactions, might bring about interesting features, such as the maximisation of selection or the switch of the selected state at stationarity. We introduce a framework, rooted in a time-scale separation analysis, which is able to capture leading non-equilibrium features using only equilibrium arguments under well-defined conditions. In particular, it is possible to identify fast-dissipation sub-networks of reactions whose Boltzmann equilibrium dominates the steady-state of the entire system as a whole. Finally, we also show that the dissipated heat (and so the entropy production) can be estimated, under some approximations, through the heat capacity of fast-dissipation sub-networks. This work provides a tool to develop an intuitive equilibrium-based grasp on complex non-isothermal reaction networks, which are important paradigms to understand the emergence of complex structures from basic building blocks.