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 mechanical state preparation II-The statistical ensemble as a thermostat

In this sequel to an earlier paper on quantal state preparation (Park & Band, 1972), the simple model used before is adapted and somewhat generalised to enable an exact treatment of the causal evolution of a system immersed in a statistical ensemble constituted of replicas of the system itself. The ensemble is initially in a state of statistical equilibrium, but the initial state of the system is arbitrary. It is established as a purely dynamical theorem that the system is eventually coerced into the same equilibrium state as that of the replicas in the ensemble. From this general result we obtain as a special case mechanical justification for the common assumption in statistical thermodynamics that an ensemble can function as a thermostat.     

Reconstructing Free Energy Profiles from Nonequilibrium Relaxation Trajectories

Reconstructing free energy profiles is an important problem in bimolecular reactions, protein folding or allosteric conformational changes. Nonequilibrium trajectories are readily measured experimentally, but their statistical significance and relation to equilibrium system properties still call for rigorous methods of assessment and interpretation. Here we introduce methods to compute the equilibrium free energy profile of a given variable from a set of short nonequilibrium trajectories, obtained by externally driving a system out of equilibrium and subsequently observing its relaxation. This protocol is not suitable for the Jarzynski equality since the irreversible work on the system is instantaneous. Assuming that the variable of interest satisfies an overdamped Langevin equation, which is frequently used for modeling biomolecular processes, we show that the trajectories sample a nonequilibrium stationary distribution that can be calculated in closed form. This allows for the estimation of the free energy via an inversion procedure that is analogous to that used in equilibrium and bypasses more complicated path integral methods, which we derive for comparison. We generalize the inversion procedure to systems with a diffusion constant that depends on the reaction coordinate, as is the case in protein folding, as well as to protocols in which the trajectories are initiated at random points. Using only a statistical pool of tens of synthetic trajectories, we demonstrate the versatility of these methods by reconstructing double and multi-well potentials, as well as a proposed profile for the hydrophobic collapse of a protein.     

Free energy in a Markovian model of a lattice spin system

A Markov process which may be thought of as a classical lattice spin system is considered. States of the system are probability measures on the configuration space, and we study the evolution of the free energy of these states with time. It is proved that for all initial states the free energy is nonincreasing and that it strictly decreases from any initial state which is shift invariant but not an equilibrium state. Finally we show that the state of the system converges weakly to the set of Gibbsian Distributions for the given interaction, and that all shift invariant equilibrium states are Gibbsian Distributions.This work was done while the author was a postdoctoral fellow in the Adolph C. and Mary Sprague Miller Institute for Basic Research in Science.     

利用扩展系综法计算水的自由能

利用扩展系综法得到了正则系综下水的TIP4P模型的自由能值为-21．485±0．035kJ/mol， 并与其他方法所得的结果作了比较.提出了选择该方法中关键参数(平衡因子)的有效方法， 并讨论了该方法的可移植性. 关键词： 自由能 TIP4P水模型 扩展系综 分子动力学模拟 水分子团     

First order phase transitions in lattice and continuous systems: Extension of Pirogov-Sinai theory

We generalize the notion of ground states in the Pirogov-Sinai theory of first order phase transitions at low temperatures, applicable to lattice systems with a finite number of periodic ground states to that of restricted ensembles with equal free energies. A restricted ensemble is a Gibbs ensemble, i.e. equilibrium probability measure, on a restricted set of configurations in the phase space of the system. When a restricted ensemble contains only one configuration it coincides with a ground state. In the more general case the entropy is also important.An example of a system we can treat by our methods is theq-state Potts model where we prove that forq sufficiently large there exists a temperature at which the system coexists inq+1 phases;q-ordered phases are small modifications of theq perfectly ordered ground states and one disordered phase which is a modification of the restricted ensemble consisting of all perfectly disordered (neighboring sites must have different spins) configurations. The free energy thus consists entirely of energy in the firstq-restricted ensembles and of entropy in the last one.Our main motivation for this work is to develop a rigorous theory for phase transitions in continuum fluids in which there is no symmetry between the phases, e.g. the liquid-vapour phase transition. The present work goes a certain way in that direction.Supported in part by NSF Grant N^r DMR81-14726-02     

Time inversion in dynamical systems

We consider the case of a dynamical system when the time evolution is generated by a nonhermitian superoperator on the states of the system. Assuming the left and right eigenvectors of this to provide complete basis sets, we propose a generalized scalar product which can be used to construct a monotonically changing functional of the state, a generalized entropy. Combining the time-dependent state with its time-reversed counterpart we can define the operation of time inversion even in this case of irreversible evolution. We require that both the forward and reversed time evolution can be obtained from a generalized action principle, and this demand serves to define the form of the time-reversed state uniquely. The work thus generalizes the quantum treatment from the unitary case to the irreversible one. We present a discussion of the approach and derive some of the direct consequences of our results.     

Derivation of the nonlinear hydrodynamic equations using multi-mode techniques

A general mode-mode coupling theory is developed for the microscopic mass, energy and momentum densities of a simple classical fluid. A projection operator method is employed to derive a generalized Langevin equation that contains nonlinearities of all orders with both convective and dissipative terms. A general nonequilibrium ensemble average, which contains local equilibrium as a special case, is employed to derive nonlinear transport equations that are nonlocal in both space and time.The nonlinear Euler and Navier-Stokes equations are recovered using a factorization procedure based on an inverse system size approximation. We show that in the context of mode-mode coupling theory, nonlinearities of all orders must be retained to derive the full nonlinear transport equations. We also slow that the space and time dependent nonequilibrium pressure and transport coefficients are functions of the nonequilibrium mass and internal energy densities. The thermodynamic closure relationships follow as a natural consequence of mode-mode coupling theory. For a system linearly displaced from equilibrium we demonstrate the role of the corrections to our factorization approximation in renormalizing the transport coefficients.     

Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences

There are only a very few known relations in statistical dynamics that are valid for systems driven arbitrarily far-from-equilibrium. One of these is the fluctuation theorem, which places conditions on the entropy production probability distribution of nonequilibrium systems. Another recently discovered far from equilibrium expression relates nonequilibrium measurements of the work done on a system to equilibrium free energy differences. In this paper, we derive a generalized version of the fluctuation theorem for stochastic, microscopically reversible dynamics. Invoking this generalized theorem provides a succinct proof of the nonequilibrium work relation.     

Length of time's arrow

An unresolved problem in physics is how the thermodynamic arrow of time arises from an underlying time reversible dynamics. We contribute to this issue by developing a measure of time-symmetry breaking, and by using the work fluctuation relations, we determine the time asymmetry of recent single molecule RNA unfolding experiments. We define time asymmetry as the Jensen-Shannon divergence between trajectory probability distributions of an experiment and its time-reversed conjugate. Among other interesting properties, the length of time's arrow bounds the average dissipation and determines the difficulty of accurately estimating free energy differences in nonequilibrium experiments.     

Canonical equilibrium distribution derived from Helmholtz potential

Plastino and Curado [A. Plastino, E.M.F. Curado, Phys. Rev. E 72 (2005) 047103] recently determined the equilibrium probability distribution for the canonical ensemble using only phenomenological thermodynamical laws as an alternative to the entropy maximization procedure of Jaynes. In the current paper we present another alternative derivation of the canonical equilibrium probability distribution, which is based on the definition of the Helmholtz free energy (and its being constant at the equilibrium) and the assumption of the uniqueness of the equilibrium probability distribution. Noting that this particular derivation is applicable for all trace-form entropies, we also apply it to the Tsallis entropy, showing that the Tsallis entropy yields genuine inverse power laws.     

Classical dissipation and asymptotic equilibrium via interaction with chaotic systems

We study the energy flow between a one-dimensional oscillator and a chaotic system with two degrees of freedom in the weak coupling limit. The oscillator's observables are averaged over an initially microcanonical ensemble of trajectories of the chaotic system, which plays the role of an environment for the oscillator. We show numerically that the oscillator's average energy exhibits irreversible dynamics and ‘thermal’ equilibrium at long times. We use linear response theory to describe the dynamics at short times and we derive a condition for the absorption or dissipation of energy by the oscillator from the chaotic system. The equilibrium properties at long times, including the average equilibrium energies and the energy distributions, are explained with the help of statistical arguments. We also check that the concept of temperature defined in terms of the ‘volume entropy’ agrees very well with these energy distributions.     

A molecular dynamics method for simulations in the canonical ensemble

A molecular dynamics simulation method which can generate configurations belonging to the canonical (T, V, N) ensemble or the constant temperature constant pressure (T, P, N) ensemble, is proposed. The physical system of interest consists of N particles (f degrees of freedom), to which an external, macroscopic variable and its conjugate momentum are added. This device allows the total energy of the physical system to fluctuate. The equilibrium distribution of the energy coincides with the canonical distribution both in momentum and in coordinate space. The method is tested for an atomic fluid (Ar) and works well.     

A hydrodynamic approach to multidimensional dissipation-based Schrödinger models from quantum Fokker-Planck dynamics

In this paper we are concerned with the modeling of quantum dissipation and diffusion effects at the level of the multidimensional Schrödinger equation. Our starting point is the quantum Fokker-Planck master equation describing dissipative interactions (of mass and energy) of the particle ensemble with a thermal bath in thermodynamic equilibrium. When considering its associated hydrodynamic system, which rules the temporal evolution of the local density and the mean fluid-flow velocity, and imposing physically admissible closure relations, these equations can be seen as describing the fluid-mechanical evolution of the macroscopic amplitude and phase of an envelope wavefunction, thus giving rise to a family of dissipative Schrödinger equations of logarithmic type whose steady state and radial dynamics are analyzed. Also, numerical comparison with the exactly solvable models for the free particle and the damped harmonic oscillator is performed.     

Computation of disordered system from the first principles of classical mechanics and ℕℙ hard problem

We study the classical 1D Heisenberg spin glasses in the framework of nearest-neighboring model. Based on the Hamilton equations we obtained the system of recurrence equations which allows to perform node-by-node calculations of a spin-chain. It is shown that calculations from the first principles of classical mechanics lead to ?? hard problem, that however in the limit of the statistical equilibrium can be calculated by ? algorithm. For the partition function of the ensemble a new representation is offered in the form of one-dimensional integral of spin-chains’ energy distribution.     

A Framework for Non-Equilibrium Statistical Ensemble Theory

Since Gibbs synthesized a general equilibrium statistical ensemble theory, many theorists have attempted to generalized the Gibbsian theory to
non-equilibrium phenomena domain, however the status of the theory of non-equilibrium phenomena can not be said as firm as well established as the
Gibbsian ensemble theory. In this work, we present a framework for the non-equilibrium statistical ensemble formalism based on a subdynamic kinetic
equation (SKE) rooted from the Brussels-Austin school and followed by some up-to-date works. The constructed key is to use a similarity transformation between Gibbsian ensembles formalism based on Liouville equation and the subdynamic ensemble formalism based on the SKE. Using this formalism, we study the spin-Boson system, as cases of weak coupling or strongly coupling, and obtain the reduced density operators for the Canonical ensembles easily.     

Fluctuations of Entropy Production in the Isokinetic Ensemble

We discuss, using computer simulation, the microscopic definition of entropy production rate in a model of a dissipative system: a sheared fluid in which the kinetic energy is kept constant via a Gaussian thermostat. The total phase space contraction rate is the sum of two statistically independent contributions: the first one is due to the work of the conservative forces, is independent of the driving force and does not vanish at zero drive, making the system nonconservative also in equilibrium. The second is due to the work of the dissipative forces, and is responsible for the average entropy production; the distribution of its fluctuations is found to verify the Fluctuation Relation of Gallavotti and Cohen. The distribution of the fluctuations of the total phase space contraction rate also verify the Fluctuation Relation. It is compared with the same quantity calculated in the isoenergetic ensemble: we find that the two ensembles are equivalent, as conjectured by many authors. Finally, we discuss the implication of our results for experiments trying to verify the validity of the FR.     

The Generalized Canonical Ensemble and Its Universal Equivalence with the Microcanonical Ensemble

This paper shows for a general class of statistical mechanical models that when the microcanonical and canonical ensembles are nonequivalent on a subset of values of the energy, there often exists a generalized canonical ensemble that satisfies a strong form of equivalence with the microcanonical ensemble that we call universal equivalence. The generalized canonical ensemble that we consider is obtained from the standard canonical ensemble by adding an exponential factor involving a continuous function g of the Hamiltonian. For example, if the microcanonical entropy is C², then universal equivalence of ensembles holds with g taken from a class of quadratic functions, giving rise to a generalized canonical ensemble known in the literature as the Gaussian ensemble. This use of functions g to obtain ensemble equivalence is a counterpart to the use of penalty functions and augmented Lagrangians in global optimization. linebreak Generalizing the paper by Ellis et al. [J. Stat. Phys. 101:999–1064 (2000)], we analyze the equivalence of the microcanonical and generalized canonical ensembles both at the level of equilibrium macrostates and at the thermodynamic level. A neat but not quite precise statement of one of our main results is that the microcanonical and generalized canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the generalized microcanonical entropy s–g is concave. This generalizes the work of Ellis et al., who basically proved that the microcanonical and canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the microcanonical entropy s is concave.