Microcanonical Energy Research Articles

Fluctuations in the entropy of Hawking radiation

We use the gravitational path integral (GPI) to compute the fluctuations of the Hawking radiation entropy around the Page curve in a two-dimensional model introduced by Penington Before the Page time, we find that δS=e−S/2, where S is the black hole entropy. This result agrees with the Haar-averaged entropy fluctuations in a bipartite system. After the Page time, we find that δS∼e−S, up to a prefactor that depends logarithmically on the width of the microcanonical energy window. This is not symmetric under exchange of subsystem sizes and so does not agree with the Haar average for a subsystem of fixed Hilbert space dimension. The discrepancy can be attributed to the fact that the black hole Hilbert space dimension is not fixed by the state preparation: even in a microcanonical ensemble with a top-hat smearing function, the GPI yields an additive fluctuation in the number of black hole states. This result, and the fact that the Page curve computed by the GPI is smooth, all point towards an ensemble interpretation of the GPI. Published by the American Physical Society 2024

Polyrate 2023: A computer program for the calculation of chemical reaction rates for polyatomics. New version announcement

Polyrate 2023: A computer program for the calculation of chemical reaction rates for polyatomics. New version announcement

Wheeler-DeWitt states of the AdS-Schwarzschild interior

We solve the Wheeler-DeWitt equation for the planar AdS-Schwarzschild interior in a minisuperspace approximation involving the volume and spatial anisotropy of the interior. A Gaussian wavepacket is constructed that is peaked on the classical interior solution. Simple observables are computed using this wavepacket, demonstrating the freedom to a choose a relational notion of ‘clock’ in the interior and characterizing the approach to the spacelike singularity. The Wheeler-DeWitt equation may be extended out through the horizon, where it describes the holographic renormalization group flow of the black hole exterior. This amounts to the Hamilton-Jacobi evolution of the metric component gtt from positive interior values to negative exterior values. The interior Gaussian wavepacket is shown to evolve into the Lorentizan partition function of the boundary conformal field theory over a microcanonical energy window.

Localization in the Discrete Non-linear Schrödinger Equation and Geometric Properties of the Microcanonical Surface

It is well known that, if the initial conditions have sufficiently high energy density, the dynamics of the classical Discrete Non-Linear Schr\"odinger Equation (DNLSE) on a lattice shows a form of breaking of ergodicity, with a finite fraction of the total charge accumulating on a few sites and residing there for times that diverge quickly in the thermodynamic limit. In this paper we show that this kind of localization can be attributed to some geometric properties of the microcanonical potential energy surface, and that it can be associated to a phase transition in the lowest eigenvalue of the Laplacian on said surface. We also show that the approximation of considering the phase space motion on the potential energy surface only, with effective decoupling of the potential and kinetic partition functions, is justified in the large connectivity limit, or fully connected model. In this model we further observe a synchronization transition, with a synchronized phase at low temperatures.

Microcanonical potential energy fluctuations and configurational density of states for nanoscale systems

In this work, we study the distribution of potential energies for the embedded atom model (EAM), commonly associated with monoatomic metals, in nanoscopic systems of the order of tens of particles. Classical molecular dynamics simulations in the microcanonical ensemble, both in solid and liquid phases, were used to produce samples of potential energy at different total energies. We characterize the shape of these fluctuations of potential energy and, in this context, we propose a model for the configurational density of states (CDOS) capable of describing the melting phase transition of a nanoscopic system. Our results show the shape of the fluctuations outside and within the range of the solid–liquid phase transition, and demonstrate the accuracy of the CDOS model for EAM systems.

Closeness of the reduced density matrix of an interacting small system to the Gibbs state.

I study the statistical description of a small quantum system, which is coupled to a large quantum environment in a generic form and with a generic interaction strength, when the total system lies in an equilibrium state described by a microcanonical ensemble. The focus is on the difference between the reduced density matrix (RDM) of the central system in this interacting case and the RDM obtained in the uncoupled case. In the eigenbasis of the central system's Hamiltonian, it is shown that the difference between diagonal elements is mainly confined by the ratio of the maximum width of the eigenfunctions of the total system in the uncoupled basis to the width of the microcanonical energy shell; meanwhile, the difference between off-diagonal elements is given by the ratio of certain property of the interaction Hamiltonian to the related level spacing of the central system. As an application, a sufficient condition is given, under which the RDM may have a canonical Gibbs form under system-environment interactions that are not necessarily weak; this Gibbs state usually includes certain averaged effect of the interaction. For central systems that interact locally with many-body quantum chaotic systems, it is shown that the RDM usually has a Gibbs form. I also study the RDM which is computed from a typical state of the total system within an energy shell.

Quantifying the Unitary Generation of Coherence from Thermal Quantum Systems.

Coherence is associated with transient quantum states; in contrast, equilibrium thermal quantum systems have no coherence. We investigate the quantum control task of generating maximum coherence from an initial thermal state employing an external field. A completely controllable Hamiltonian is assumed allowing the generation of all possible unitary transformations. Optimizing the unitary control to achieve maximum coherence leads to a micro-canonical energy distribution on the diagonal energy representation. We demonstrate such a control scenario starting from a given Hamiltonian applying an external field, reaching the control target. Such an optimization task is found to be trap-less. By constraining the amount of energy invested by the control, maximum coherence leads to a canonical energy population distribution. When the optimization procedure constrains the final energy too tightly, local suboptimal traps are found. The global optimum is obtained when a small Lagrange multiplier is employed to constrain the final energy. Finally, we explore the task of generating coherences restricted to be close to the diagonal of the density matrix in the energy representation.

Dynamical Typicality Approach to Eigenstate Thermalization.

We consider the set of all initial states within a microcanonical energy shell of an isolated many-body quantum system, which exhibit an arbitrary but fixed nonequilibrium expectation value for some given observable A. On the condition that this set is not too small, it is shown by means of a dynamical typicality approach that most such initial states exhibit thermalization if and only if A satisfies the so-called weak eigenstate thermalization hypothesis (wETH). Here, thermalization means that the expectation value of A spends most of its time close to the microcanonical value after initial transients have died out. The wETH means that, within the energy shell, most eigenstates of the pertinent system Hamiltonian exhibit very similar expectation values of A.

Numerical Large Deviation Analysis of the Eigenstate Thermalization Hypothesis.

A plausible mechanism of thermalization in isolated quantum systems is based on the strong version of the eigenstate thermalization hypothesis (ETH), which states that all the energy eigenstates in the microcanonical energy shell have thermal properties. We numerically investigate the ETH by focusing on the large deviation property, which directly evaluates the ratio of athermal energy eigenstates in the energy shell. As a consequence, we have systematically confirmed that the strong ETH is indeed true even for near-integrable systems. Furthermore, we found that the finite-size scaling of the ratio of athermal eigenstates is a double exponential for nonintegrable systems. Our result illuminates the universal behavior of quantum chaos, and suggests that a large deviation analysis would serve as a powerful method to investigate thermalization in the presence of the large finite-size effect.

An $L^{p}$ theory of sparse graph convergence II: LD convergence, quotients and right convergence

We extend the $L^{p}$ theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence and large deviation convergence. Our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree, while the proofs require new techniques based on uniform upper regularity. Examples to which our theory applies include stochastic block models, power law graphs and sparse versions of $W$-random graphs.

Many-body localization and thermalization: Insights from the entanglement spectrum

We study the entanglement spectrum in the many body localizing and thermalizing phases of one and two dimensional Hamiltonian systems, and periodically driven `Floquet' systems. We focus on the level statistics of the entanglement spectrum as obtained through numerical diagonalization, finding structure beyond that revealed by more limited measures such as entanglement entropy. In the thermalizing phase the entanglement spectrum obeys level statistics governed by an appropriate random matrix ensemble. For Hamiltonian systems this can be viewed as evidence in favor of a strong version of the eigenstate thermalization hypothesis (ETH). Similar results are also obtained for Floquet systems, where they constitute a result `beyond ETH', and show that the corrections to ETH governing the Floquet entanglement spectrum have statistical properties governed by a random matrix ensemble. The particular random matrix ensemble governing the Floquet entanglement spectrum depends on the symmetries of the Floquet drive, and therefore can depend on the choice of origin of time. In the many body localized phase the entanglement spectrum is also found to show level repulsion, following a semi-Poisson distribution (in contrast to the energy spectrum, which follows a Poisson distribution). This semi-Poisson distribution is found to come mainly from states at high entanglement energies. The observed level repulsion only occurs for interacting localized phases. We also demonstrate that equivalent results can be obtained by calculating with a single typical eigenstate, or by averaging over a microcanonical energy window - a surprising result in the localized phase. This discovery of new structure in the pattern of entanglement of localized and thermalizing phases may open up new lines of attack on many body localization, thermalization, and the localization transition.

Typicality of Thermal Equilibrium and Thermalization in Isolated Macroscopic Quantum Systems

Based on the view that thermal equilibrium should be characterized through macroscopic observations, we develop a general theory about typicality of thermal equilibrium and the approach to thermal equilibrium in macroscopic quantum systems. We first formulate the notion that a pure state in an isolated quantum system represents thermal equilibrium. Then by assuming, or proving in certain classes of nontrivial models (including that of two bodies in thermal contact), large-deviation type bounds (which we call thermodynamic bounds) for the microcanonical ensemble, we prove that to represent thermal equilibrium is a typical property for pure states in the microcanonical energy shell. We believe that the typicality, along with the empirical success of statistical mechanics, provides a sound justification of equilibrium statistical mechanics. We also establish the approach to thermal equilibrium under two different assumptions; one is that the initial state has a moderate energy distribution, and the other is the energy eigenstate thermalization hypothesis.

VACANCY GENERATION AND ADSORPTION OF Cu ATOM AT Ag(1 1 1) SURFACE DURING DIFFUSION OF Cu-TRIMER

In this paper, the vacancy generation at (1 1 1) surface of silver ( Ag ) and the adsorption of copper ( Cu ) atom in the vacant site at Ag (1 1 1) surface has been studied using molecular dynamics (MD) simulation technique. The run of micro-canonical constant energy (NVE) ensemble results the generation of vacancy at 700 K in the presence of Cu -trimer, and its migration is observed toward the island as a result of self-diffusion of the surface layer atom. The energy of vacancy generation at Ag surface is found to be Ev = 1.078 eV , in the presence of Cu -trimer. The popped-up Ag adatom combines with Cu -trimer, making a mixed-tetramer ( Cu 3– Ag island). The adsorption energy of the Cu -atom among the mixed-tetramer island into the substitutional site is Es = ~ -0.813 eV , leaving a mixed-trimer ( Cu 2– Ag island) at Ag (1 1 1). Moreover, the presence of the island creates some anharmonic effects like dislocations and fissures at the surface. These dislocations and fissures show a trend of attraction to each other and also migrate toward the island, during the time evolution of the system.

The molecular dynamic study of anharmonic effects at Cu(111) and Ag(111) surfaces in the presence of Cu- and Ag-trimer island

The molecular dynamics (MD) technique based on semi-empirical potentials, is used to carry out the diffusion of Cu- and Ag-trimer on Cu- and Ag(111) surface at 300, 500 and 700 K temperatures. The constant energy MD simulation elaborates the anharmonic effects at the surface such as fissures, dislocations and vacancy creation, in the presence of island. The fissures and dislocations formed are in the range of 1.5–4 Å and 1–7 Å, respectively, from the island's position. The Cu and Ag islands both diffuse easily on Cu(111) surface, manipulate that the trend of diffusion is faster on Cu surface as compared to Ag surface. The process of breaking and opening of the island has also been observed. Moreover, a surface atom popped-up at 700 K by creating a vacancy near the Cu island on Ag surface. The rate of diffusion increases with the increase in temperature, both for homo- and hetero-cases. • The micro-canonical constant energy run results the vacancy creation at 700 K in the presence of Cu trimer. • The presence of the Cu 3 island creates some anharmonic effects like dislocations and fissures at the Ag(111) surface. • The trend of diffusion is faster on Cu surface as compared to Ag surface for Cu and Ag trimer. • The rate of diffusion increases with the increase in temperature.

Thermalization in the two-body random ensemble

Using the ergodicity principle for the expectation values of several types of observables, weinvestigate the thermalization process in isolated fermionic systems. These are described bythe two-body random ensemble, which is a paradigmatic model to study quantum chaosand especially the dynamical transition from integrability to chaos. By means of exactdiagonalizations we analyze the relevance of the eigenstate thermalization hypothesis aswell as the influence of other factors, such as the energy and structure of the initial state,or the dimension of the Hilbert space. We also obtain analytical expressions linkingthe degree of thermalization for a given observable with the so-called number ofprincipal components for transition strengths originating at a given energy, with thedimensions of the whole Hilbert space and microcanonical energy shell, and with thecorrelations generated by the observable. As the strength of the residual interaction isincreased, an order-to-chaos transition takes place, and we show that the onset ofWigner spectral fluctuations, which is the standard signature of chaos, is notsufficient to guarantee thermalization in finite systems. When all the signatures ofchaos are fulfilled, including the quasicomplete delocalization of eigenfunctions,the eigenstate thermalization hypothesis is the mechanism responsible for thethermalization of certain types of observables, such as (linear combinations of)occupancies and strength function operators. Our results also suggest that fullychaotic systems will thermalize relative to most observables in the thermodynamiclimit.

Long-time behavior of macroscopic quantum systems

The renewed interest in the foundations of quantum statistical mechanics in recent years has led us to study John von Neumann's 1929 article on the quantum ergodic theorem. We have found this almost forgotten article, which until now has been available only in German, to be a treasure chest, and to be much misunderstood. In it, von Neumann studied the long-time behavior of macroscopic quantum systems. While one of the two theorems announced in his title, the one he calls the "quantum H-theorem", is actually a much weaker statement than Boltzmann's classical H-theorem, the other theorem, which he calls the "quantum ergodic theorem", is a beautiful and very non-trivial result. It expresses a fact we call "normal typicality" and can be summarized as follows: For a "typical" finite family of commuting macroscopic observables, every initial wave function $\psi_0$ from a micro-canonical energy shell so evolves that for most times $t$ in the long run, the joint probability distribution of these observables obtained from $\psi_t$ is close to their micro-canonical distribution.

The extended Gaussian ensemble and metastabilities in the Blume–Capel model

The Blume-Capel model with infinite-range interactions presents analytical solutions in both canonical and microcanonical ensembles and therefore, its phase diagram is known in both ensembles. This model exhibits nonequivalent solutions and the microcanonical thermodynamical features present peculiar behaviors like nonconcave entropy, negative specific heat, and a jump in the thermodynamical temperature. Examples of nonequivalent ensembles are in general related to systems with long-range interactions that undergo canonical first-order phase transitions. Recently, the extended gaussian ensemble (EGE) solution was obtained for this model. The gaussian ensemble and its extended version can be considered as a regularization of the microcanonical ensemble. They are known to play the role of an interpolating ensemble between the microcanonical and the canonical ones. Here, we explicitly show how the microcanonical energy equilibrium states related to the metastable and unstable canonical solutions for the Blume-Capel model are recovered from EGE, which presents a concave "extended" entropy as a function of energy.

Thermalization and its mechanism for generic isolated quantum systems

An understanding of the temporal evolution of isolated many-body quantum systems has long been elusive. Recently, meaningful experimental studies of the problem have become possible, stimulating theoretical interest. In generic isolated systems, non-equilibrium dynamics is expected to result in thermalization: a relaxation to states in which the values of macroscopic quantities are stationary, universal with respect to widely differing initial conditions, and predictable using statistical mechanics. However, it is not obvious what feature of many-body quantum mechanics makes quantum thermalization possible in a sense analogous to that in which dynamical chaos makes classical thermalization possible. For example, dynamical chaos itself cannot occur in an isolated quantum system, in which the time evolution is linear and the spectrum is discrete. Some recent studies even suggest that statistical mechanics may give incorrect predictions for the outcomes of relaxation in such systems. Here we demonstrate that a generic isolated quantum many-body system does relax to a state well described by the standard statistical-mechanical prescription. Moreover, we show that time evolution itself plays a merely auxiliary role in relaxation, and that thermalization instead happens at the level of individual eigenstates, as first proposed by Deutsch and Srednicki. A striking consequence of this eigenstate-thermalization scenario, confirmed for our system, is that knowledge of a single many-body eigenstate is sufficient to compute thermal averages-any eigenstate in the microcanonical energy window will do, because they all give the same result.

Negative specific heat, phase transition and particles spilling from a potential well

For a finite number of noninteracting particles in a box with a potential well in the center, the microcanonical kinetic energy in dependence on the total energy as it is negative can be classified into three categories. The first exhibits a monotonical rise and the specific heat is positive. The second shows a diminishing sawtooth wave with a global rise. The last corresponds to the extreme case and takes the regular sawtooth wave form. The sawtooth wave portion associates periodically a kinetic energy fall in spite of an increase of the total energy; and we attribute to such a fall the negative specific heat. The phase transition can be defined when the relatively dense particle state in the well and relatively dilute particle state in the rest volume of the box coexist, and the appearance of the negative specific heat is sufficient but not necessary for the onset of the phase transition.

Multiplicity fluctuations in relativistic nuclear collisions: Statistical model versus experimental data

The multiplicity distributions of hadrons produced in central nucleus-nucleus collisions are studied within the hadron-resonance gas model in the large-volume limit. The microscopic correlator method is used to enforce conservation of three charges---baryon number, electric charge, and strangeness---in the canonical ensemble. In addition, in the microcanonical ensemble energy conservation is included. An analytical method is used to account for resonance decays. The multiplicity distributions and the scaled variances for negatively, positively, and all charged hadrons are calculated along the chemical freeze-out line of central $\mathrm{Pb}+\mathrm{Pb}$ ($\mathrm{Au}+\mathrm{Au}$) collisions from GSI Schwerionen Synchrotron to CERN Large Hadron Collider energies. Predictions obtained within different statistical ensembles are compared with the preliminary NA49 experimental results on central $\mathrm{Pb}+\mathrm{Pb}$ collisions in the SPS energy range. The measured fluctuations are significantly narrower than the Poisson ones and clearly favor expectations for the microcanonical ensemble. Thus this is the first observation of the recently predicted suppression of the multiplicity fluctuations in relativistic gases in the thermodynamical limit owing to conservation laws.