核温度的动力学计算

介绍了测量热力学微正则系综中哈密顿动力学系统温度的一种新的动力学途径,即在各态历经性的假设下,温度可作为函数（Ｈ／‖Ｈ‖２）在能量面上的时间平均算出．这一方法不仅给出了确定温度的一种有效的计算途径,而且也提供了动力学系统理论和哈密顿系统的统计力学之间的内在联系． A new dynamical approach for measuring the temperature of a Hamiltonian dynamical system in the microcanonical ensemble of thermodynamics is presented. It shows that under the hypothesis of ergodicity the temperature can be computed as a time average of the function, ·(H/‖H‖ 2) , on the energy surface. This method not only yields an efficient computational approach for determining the temperature, but also provides an intrinsic link between dynamical system theory and the statistical...     

Combinatorial and Topological Analysis of the Ising Chain in a Field

We present an alternative solution of the Ising chain in a field under free and periodic boundary conditions, in the microcanonical, canonical, and grand canonical ensembles, from a unified combinatorial and topological perspective. In particular, the computation of the per-site entropy as a function of the energy unveils a residual value for critical values of the magnetic field, a phenomenon for which we provide a topological interpretation and a connection with the Fibonacci sequence. We also show that, in the thermodynamic limit, the per-site microcanonical entropy is equal to the logarithm of the per-site Euler characteristic. The canonical and grand canonical partition functions are identified as combinatorial generating functions of the microcanonical problem, which allows us to evaluate them. A detailed analysis of the magnetic field-dependent thermodynamics, including positive and negative temperatures, reveals interesting features. Finally, we emphasize that our combinatorial approach to the canonical ensemble allows exact computation of the thermally averaged value <????> of the Euler characteristic associated with the spin configurations of the chain, which is discontinuous at the critical fields, and whose thermal behavior is expected to determine the phase transition of the model. Indeed, our results show that the conjecture <????>?(T _C)?=?0, where T _C is the critical temperature, is valid for the Ising chain.     

Universal canonical entropy for gravitating systems

The thermodynamics of general relativistic systems with boundary, obeying a Hamiltonian constraint in the bulk, is determined solely by the boundary quantum dynamics, and hence by the area spectrum. Assuming, for large area of the boundary, (a) an area spectrum as determined by non-perturbative canonical quantum general relativity (NCQGR), (b) an energy spectrum that bears a power law relation to the area spectrum, (c) an area law for the leading order microcanonical entropy, leading thermal fluctuation corrections to the canonical entropy are shown to be logarithmic in area with a universal coefficient. Since the microcanonical entropy also has universal logarithmic corrections to the area law (from quantum space-time fluctuations, as found earlier) the canonical entropy then has a universal form including logarithmic corrections to the area law. This form is shown to be independent of the index appearing in assumption (b). The index, however, is crucial in ascertaining the domain of validity of our approach based on thermal equilibrium.     

Microscopic diagonal entropy and its connection to basic thermodynamic relations

We define a diagonal entropy (d-entropy) for an arbitrary Hamiltonian system as S_d=-∑_nρ_nnlnρ_nn with the sum taken over the basis of instantaneous energy states. In equilibrium this entropy coincides with the conventional von Neumann entropy S_n = −Trρ ln ρ. However, in contrast to S_n, the d-entropy is not conserved in time in closed Hamiltonian systems. If the system is initially in stationary state then in accord with the second law of thermodynamics the d-entropy can only increase or stay the same. We also show that the d-entropy can be expressed through the energy distribution function and thus it is measurable, at least in principle. Under very generic assumptions of the locality of the Hamiltonian and non-integrability the d-entropy becomes a unique function of the average energy in large systems and automatically satisfies the fundamental thermodynamic relation. This relation reduces to the first law of thermodynamics for quasi-static processes. The d-entropy is also automatically conserved for adiabatic processes. We illustrate our results with explicit examples and show that S_d behaves consistently with expectations from thermodynamics.     

Thermodynamics of the HMF model with a magnetic field

We study the thermodynamics of the Hamiltonian mean field (HMF) model with an external potential playing the role of a “magnetic field”. If we consider only fully stable states, the caloric curve does not present any phase transition. However, if we take into account metastable states (for a restricted class of perturbations), we find a very rich phenomenology. In particular, the caloric curve displays a region of negative specific heat in the microcanonical ensemble in which the temperature decreases as the energy increases. This leads to ensembles inequivalence and to zeroth order phase transitions similar to the “gravothermal catastrophe” and to the “isothermal collapse” of self-gravitating systems. In the present case, they correspond to the reorganization of the system from an “anti-aligned” phase (magnetization pointing in the direction opposite to the magnetic field) to an “aligned” phase (magnetization pointing in the same direction as the magnetic field). We also find that the magnetic susceptibility can be negative in the microcanonical ensemble so that the magnetization decreases as the magnetic field increases. The magnetic curves can take various shapes depending on the values of energy or temperature. We describe first order phase transitions and hysteretic cycles involving positive or negative susceptibilities. We also show that this model exhibits gaps in the magnetization at fixed energy, resulting in ergodicity breaking.     

Effective spin-glass Hamiltonian for the anomalous dynamics of the HMF model

We discuss an effective spin-glass Hamiltonian which can be used to study the glassy-like dynamics observed in the metastable states of the Hamiltonian mean field (HMF) model. By means of the Replica formalism, we were able to find a self-consistent equation for the glassy order parameter which reproduces, in a restricted energy region below the phase transition, the microcanonical simulations for the polarization order parameter recently introduced in the HMF model.     

相对论热力学的哈密顿函数

Planck-Einstein的相对论热力学理论一直存在着争议,特别是它是否遵守动量－能量守恒定律,本文从热力学的Lagrange函数,导出了对应的Hamilton函数,Hamilton函数的意义极为明确,它包含了环境对体系作用的耦合能量,在一般情况下,不是体系的能量,而是Hamilton量,和动量组成了四维矢量,这不仅证明了P－E理论是符合守恒定律的,而且扩展了这一理论。     

Violation of the zeroth law of thermodynamics in systems with negative specific heat

We show that systems with negative specific heat can violate the zeroth law of thermodynamics. By both numerical simulations and by using exact expressions for free energy and microcanonical entropy, it is shown that if two systems with the same intensive parameters but with negative specific heat are thermally coupled, they undergo a process in which the total entropy increases irreversibly. The final equilibrium is such that two phases appear; that is, the subsystems have different magnetizations and internal energies at temperatures which are equal in both systems, but that can be different from the initial temperature.     

Is computation reversible?

Recent studies have suggested that computation is essentially reversible, provided no information is lost. This is a consequence of Landauer’s principle which only requires energy expenditure and entropy increase for information deletion. In this paper we propose to treat information as being intrinsic to points of non-analyticity, so that the movement of information is always associated with the dissipation of heat. This allows us to construct a theory consistent with causality, and the second law of thermodynamics. Since computation requires the movement of information bits through finite volume gates, energy is dissipated even when information is not destroyed, thus indicating that computation is fundamentally non-reversible.     

A one parameter fit for glassy dynamics as a quantum corollary of the liquid to solid transition

We apply microcanonical ensemble considerations to suggest that, whenever it may thermalise, a general disorder-free many-body Hamiltonian of a typical atomic system has solid-like eigenstates at low energies and fluid-type (and gaseous, plasma) eigenstates associated with energy densities exceeding those present in the melting (and, respectively, higher energy) transition(s). In particular, the lowest energy density at which the eigenstates of such a clean many body atomic system undergo a non-analytic change is that of the melting (or freezing) transition. We invoke this observation to analyse the evolution of a liquid upon supercooling (i.e. cooling rapidly enough to avoid solidification below the freezing temperature). Expanding the wavefunction of a supercooled liquid in the complete eigenbasis of the many-body Hamiltonian, only the higher energy liquid-type eigenstates contribute significantly to measurable hydrodynamic relaxations (e.g. those probed by viscosity) while static thermodynamic observables become weighted averages over both solid- and liquid-type eigenstates. Consequently, when extrapolated to low temperatures, hydrodynamic relaxation times of deeply supercooled liquids (i.e. glasses) may seem to diverge at nearly the same temperature at which the extrapolated entropy of the supercooled liquid becomes that of the solid. In this formal quantum framework, the increasingly sluggish (and spatially heterogeneous) dynamics in supercooled liquids as their temperature is lowered stems from the existence of the single non-analytic change of the eigenstates of the clean many-body Hamiltonian at the equilibrium melting transition present in low energy solid-type eigenstates. We derive a single (possibly computable) dimensionless parameter fit to the viscosity and suggest other testable predictions of our approach.     

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.     

Fast algorithm for the simulation of Ising models

Using a new microcanonical algorithm efficiently vectorized on a Cray XMP, we reach a simulation speed of 1.5 nsec per update of one spin, three times faster than the best previous method known to us. Data for the nonlinear relaxation with conserved energy are presented for the two-dimensional Ising model.     

Isolated and adiabatic susceptibilities

It is shown from quantum statistical calculations that the possible difference between isolated and adiabatic susceptibilities is due to a partial energy being conserved in a slowly increasing field. This energy is constructed using the unitary space of linear operators. The consequences for thermodynamics are discussed, and the isolated susceptibility is found to be exactly equal to the total adiabatic susceptibility as obtained from generalized thermodynamics (Fick and Schwegler) with two different temperatures.     

Nonthermal statistics in isolated quantum spin clusters after a series of perturbations

We show numerically that a finite isolated cluster of interacting spins 1/2 exhibits a surprising nonthermal statistics when subjected to a series of small nonadiabatic perturbations by an external magnetic field. The resulting occupations of energy eigenstates are significantly higher than the thermal ones on both the low and the high ends of the energy spectra. This behavior semiquantitatively agrees with the statistics predicted for the so-called "quantum microcanonical" ensemble, which includes all possible quantum superpositions with a given energy expectation value. Our findings also indicate that the eigenstates of the perturbation operators are generically localized in the energy basis of the unperturbed Hamiltonian. This kind of localization possibly protects the thermal behavior in the macroscopic limit.     

Conserved charges and thermodynamics of the spinning Gödel black hole

We compute the mass, angular momenta, and charge of the G?del-type rotating black hole solution to five-dimensional minimal supergravity. A generalized Smarr formula is derived, and the first law of thermodynamics is verified. The computation rests on a new approach to conserved charges in gauge theories that allows for their computation at finite radius.     

Quantum mechanical hamiltonian models of turing machines

Quantum mechanical Hamiltonian models, which represent an aribtrary but finite number of steps of any Turing machine computation, are constructed here on a finite lattice of spin-1/2 systems. Different regions of the lattice correspond to different components of the Turing machine (plus recording system). Successive states of any machine computation are represented in the model by spin configuration states. Both time-independent and time-dependent Hamiltonian models are constructed here. The time-independent models do not dissipate energy or degrade the system state as they evolve. They operate close to the quantum limit in that the total system energy uncertainty/computation speed is close to the limit given by the time-energy uncertainty relation. However, the model evolution is time global and the Hamiltonian is more complex. The time-dependent models do not degrade the system state. Also they are time local and the Hamiltonian is less complex.     

Microcanonical versus canonical analysis of protein folding

The microcanonical analysis is shown to be a powerful tool to characterize the protein folding transition and to neatly distinguish between good and bad folders. An off-lattice model with parameter chosen to represent polymers of these two types is used to illustrate this approach. Both canonical and microcanonical ensembles are employed. The required calculations were performed using parallel tempering Monte Carlo simulations. The most revealing features of the folding transition are related to its first-order-like character, namely, the S-bend pattern in the caloric curve, which gives rise to negative microcanonical specific heats, and the bimodality of the energy distribution function at the transition temperatures. Models for a good folder are shown to be quite robust against perturbations in the interaction potential parameters.     

Sampling along reaction coordinates with the Wang-Landau method

The multiple range random walk algorithm recently proposed by Wang and Landau [2001, Phys. Rev. Lett., 86, 2050] is adapted to the computation of free energy profiles for molecular systems along reaction coordinates. More generally, we show how to extract partial averages in various statistical ensembles without invoking simulations with constraints, biasing potentials or unknown parameters. The method is illustrated on a model 10-dimensional potential energy surface, for which analytical results are obtained. It is then applied to the potential of mean force associated with the dihedral angle of the butane molecule in the gas phase and in carbon tetrachloride solvent. Finally, isomerization in a small rocksalt cluster, (NaF)₄, is investigated in the microcanonical ensemble, and the results are compared to those of parallel tempering Monte Carlo.     

On correlation function methods for detecting the stochastic transition

The use of correlation function methods to predict the onset of chaotic motion in conservative Hamiltonian systems is critically examined. It is shown that microcanonical correlation functions do not, in general, provide a convenient criterion to distinguish between stable and unstable motions.