Sharp Convergence Rate of the Glimm Scheme for General Nonlinear Hyperbolic Systems

Consider a general strictly hyperbolic, quasilinear system, in one space dimesion

Thermodynamics of Currents in Nonequilibrium Diffusive Systems: Theory and Simulation

Understanding the physics of nonequilibrium systems remains as one of the major challenges of modern theoretical physics. We believe nowadays that this problem can be cracked in part by investigating the macroscopic fluctuations of the currents characterizing nonequilibrium behavior, their statistics, associated structures and microscopic origin. This fundamental line of research has been severely hampered by the overwhelming complexity of this problem. However, during the last years two new powerful and general methods have appeared to investigate fluctuating behavior that are changing radically our understanding of nonequilibrium physics: a powerful macroscopic fluctuation theory (MFT) and a set of advanced computational techniques to measure rare events. In this work we study the statistics of current fluctuations in nonequilibrium diffusive systems, using macroscopic fluctuation theory as theoretical framework, and advanced Monte Carlo simulations of several stochastic lattice gases as a laboratory to test the emerging picture. Our quest will bring us from (1) the confirmation of an additivity conjecture in one and two dimensions, which considerably simplifies the MFT complex variational problem to compute the thermodynamics of currents, to (2) the discovery of novel isometric fluctuation relations, which opens an unexplored route toward a deeper understanding of nonequilibrium physics by bringing symmetry principles to the realm of fluctuations, and to (3) the observation of coherent structures in fluctuations, which appear via dynamic phase transitions involving a spontaneous symmetry breaking event at the fluctuating level. The clear-cut observation, measurement and characterization of these unexpected phenomena, well described by MFT, strongly support this theoretical scheme as the natural theory to understand the thermodynamics of currents in nonequilibrium diffusive media, opening new avenues of research in nonequilibrium physics.     

Stochastic Theory of Nucleation in Nonequilibrium Bistable Reaction Systems

Nucleation is one of the fundamental types of forming stable dissipative nonequilibrium structures in metastable initial states. The nucleation process of a bistable chemical reaction system is investigated. First a microdroplet with radius R₀ of the new state imbedded in the parent one is formed by a local particle number fluctuation. The radius of the microdroplet R(t) fluctuates due to the interference of reaction and diffusion. If R(t) considerably exceeds a critical value R_k the droplet starts to grow deterministically. The mean nucleation time of one droplet in weakly supersaturated systems is calculated using the concept of the mean first passage time.     

Thermodynamic Formalism for Systems with Markov Dynamics

The thermodynamic formalism allows one to access the chaotic properties of equilibrium and out-of-equilibrium systems, by deriving those from a dynamical partition function. The definition that has been given for this partition function within the framework of discrete time Markov chains was not suitable for continuous time Markov dynamics. Here we propose another interpretation of the definition that allows us to apply the thermodynamic formalism to continuous time. We also generalize the formalism—a dynamical Gibbs ensemble construction—to a whole family of observables and their associated large deviation functions. This allows us to make the connection between the thermodynamic formalism and the observable involved in the much-studied fluctuation theorem. We illustrate our approach on various physical systems: random walks, exclusion processes, an Ising model and the contact process. In the latter cases, we identify a signature of the occurrence of dynamical phase transitions. We show that this signature can already be unraveled using the simplest dynamical ensemble one could define, based on the number of configuration changes a system has undergone over an asymptotically large time window.     

Mathematical Constraints in Multiscale Subgrid-Scale Modeling of Nonlinear Systems

To shed light on the subgrid-scale(SGS)modeling methodology of nonlinear systems such as the Navier—Stokes turbulence,we define the concepts of assumption and restriction in the modeling procedure,which are shown by generalized derivation of three general mathematical constraints for different combinations of restrictions.These constraints are verified numerically in a one-dimensional nonlinear advection equation.This study is expected to inspire future research on the SGS modeling methodology of nonlinear systems.     

Analysis of the Reaction Rate Coefficients for Slow Bimolecular Chemical Reactions

Simple bimolecular reactions $A_1+A_2\rightleftharpoons A_3+A_4$ are analyzed within the framework of the Boltzmann equation in the initial stage of a chemical reaction with the system far from chemical equilibrium. The Chapman-Enskog methodology is applied to determine the coefficients of the expansion of the distribution functions in terms of Sonine polynomials for peculiar molecular velocities. The results are applied to the reaction $H_2 +Cl\rightleftharpoons HCl+H$ , and the influence of the non-Maxwellian distribution and of the activation-energy dependent reactive cross sections upon the forward and reverse reaction rate coefficients are discussed.     

含双时间步法的化学非平衡流解耦算法

发展基于隐式双时间步法的化学非平衡流解耦型计算方法.采用算子分裂法对流动和反应进行解耦处理,流动方程组通过双时间步方法求解;源项方程组采用二阶梯形公式迭代求解;提出"源项消去"法,以消除化学反应源项对流动求解引入的误差,从而保证流动方程组求解的时间精度.理论分析和计算结果表明,方法既可以保证双时间步法的求解效率,又可以获得比较精确的非定常计算结果.     

Computation of Current Cumulants for Small Nonequilibrium Systems

We analyze a systematic algorithm for the exact computation of the current cumulants in stochastic nonequilibrium systems, recently discussed in the framework of full counting statistics for mesoscopic systems. This method is based on identifying the current cumulants from a Rayleigh-Schrödinger perturbation expansion for the generating function. Here it is derived from a simple path-distribution identity and extended to the joint statistics of multiple currents. For a possible thermodynamical interpretation, we compare this approach to a generalized Onsager-Machlup formalism. We present calculations for a boundary driven Kawasaki dynamics on a one-dimensional chain, both for attractive and repulsive particle interactions.     

A Unified Mathematical Formalism for the Dirac Formulation of Quantum Mechanics

We revise the mathematical implementation of the Dirac formulation of quantum mechanics, presenting a rigorous framework that unifies most of versions of this implementation.     

从功能定理到热力学第一定律到普遍的能量守恒定律

着重阐明功和能的根本意义，在质点的功能定理和质心系的基础上说明摩擦生热及热力学第一定律的意义，最后说明普遍的能量守恒定律的建立过程及其意义。     

通过类比Rice-Ramsperger-Kassel-Marcus理论再议Marcus的电子转移速率：关于线性和非线性溶剂化情形的统一形式

R. A. Marcus在他开拓性的工作中,考察了溶剂化效应对电子转移过程的影响,并给出了著名的非绝热电子转移速率公式. 本文基于热力学溶剂化势能面的分析,从Rice-Ramsperger-Kassel-Marcus理论的角度重新考察了Marcus的公式. 由类比Rice-Ramsperger-Kassel-Marcus得到的理论,不仅可以适用于线性溶剂化的情形并得到Marcus的速率公式,也同样可以用于非线性溶剂化的情形. 在非线性溶剂化的情形下,会存在溶剂化势能面的多点交叉. 本文平行地考察了Fermi黄金规则给出的相应结果,并对比本工作中所提出的Rice-Ramsperger-Kassel-Marcus类似理论进行了批判性的讨论. 作为例释,考察了二次型溶剂化的情形. 对于这种情形,物理上存在良好的描述方案.     

A Review of the System-Intrinsic Nonequilibrium Thermodynamics in Extended Space (MNEQT) with Applications

The review deals with a novel approach (MNEQT) to nonequilibrium thermodynamics (NEQT) that is based on the concept of internal equilibrium (IEQ) in an enlarged state space SZ involving internal variables as additional state variables. The IEQ macrostates are unique in SZ and have no memory just as EQ macrostates are in the EQ state space SX⊂SZ. The approach provides a clear strategy to identify the internal variables for any model through several examples. The MNEQT deals directly with system-intrinsic quantities, which are very useful as they fully describe irreversibility. Because of this, MNEQT solves a long-standing problem in NEQT of identifying a unique global temperature T of a system, thus fulfilling Planck’s dream of a global temperature for any system, even if it is not uniform such as when it is driven between two heat baths; T has the conventional interpretation of satisfying the Clausius statement that the exchange macroheatdeQflows from hot to cold, and other sensible criteria expected of a temperature. The concept of the generalized macroheat dQ=deQ+diQ converts the Clausius inequality dS≥deQ/T0 for a system in a medium at temperature T0 into the Clausius equalitydS≡dQ/T, which also covers macrostates with memory, and follows from the extensivity property. The equality also holds for a NEQ isolated system. The novel approach is extremely useful as it also works when no internal state variables are used to study nonunique macrostates in the EQ state space SX at the expense of explicit time dependence in the entropy that gives rise to memory effects. To show the usefulness of the novel approach, we give several examples such as irreversible Carnot cycle, friction and Brownian motion, the free expansion, etc.     

Refined Second Law of Thermodynamics for Fast Random Processes

We establish a refined version of the Second Law of Thermodynamics for Langevin stochastic processes describing mesoscopic systems driven by conservative or non-conservative forces and interacting with thermal noise. The refinement is based on the Monge-Kantorovich optimal mass transport and becomes relevant for processes far from quasi-stationary regime. General discussion is illustrated by numerical analysis of the optimal memory erasure protocol for a model for micron-size particle manipulated by optical tweezers.     

General Formalism for Mapping of Two-Mode Classical Canonical Transformations to Quantum Unitary operators

Two types of canonical transformations in two-mode classical phase space are mapped into the quantum mechanical Hilbert space to produce some new normally ordered unitary operators. These operators are evaluated in the coordinate (momentum) representations using the "integration within ordered product technique, and the mapping is maniferrtly apparent in the derivation. New generalixed coherent states are constructed in terms of these operators, and the uncertainty relations for these states are analysed.     

Nonequilibrium Measurements of Free Energy Differences for Microscopically Reversible Markovian Systems

An equality has recently been shown relating the free energy difference between two equilibrium ensembles of a system and an ensemble average of the work required to switch between these two configurations. In the present paper it is shown that this result can be derived under the assumption that the system's dynamics is Markovian and microscopically reversible.     

Quantum Fisher Information Gap for Systems with Nonlinear Hamiltonians

Quantum Fisher information(QFI) gap characterizes the stability of QFI to space directions. We study the QFI distributions and QFI gap for quantum states generated from nonlinear Hamiltonians for both spin and bosonic systems. We find that the same spin-squeezing parameter(or principle squeezing parameter) corresponds to two different values QFI gap, and the locations of all extreme points of the QFI are explicitly given.