arXiv subjects

Joel L. Lebowitz

Publications and source records attributed to Joel L. Lebowitz.

At least 37 records · Page 2Linked to original sources

arXiv2019

Gibbs and Boltzmann Entropy in Classical and Quantum Mechanics

The Gibbs entropy of a macroscopic classical system is a function of a probability distribution over phase space, i.e., of an ensemble. In contrast, the Boltzmann entropy is a function on phase space, and is thus defined for an individual system. Our aim is to discuss and compare these two notions of entropy, along with the associated ensemblist and individualist views of thermal equilibrium. Using the Gibbsian ensembles for the computation of the Gibbs entropy, the two notions yield the same (leading order) values for the entropy of a macroscopic system in thermal equilibrium. The two approaches do not, however, necessarily agree for non-equilibrium systems. For those, we argue that the Boltzmann entropy is the one that corresponds to thermodynamic entropy, in particular in connection with the second law of thermodynamics. Moreover, we describe the quantum analog of the Boltzmann entropy, and we argue that the individualist (Boltzmannian) concept of equilibrium is supported by the recent works on thermalization of closed quantum systems.

cond-mat.stat-mech↗

arXiv2019

Diffusive limit for a Boltzmann-like equation with non-conserved momentum

We consider a kinetic model whose evolution is described by a Boltzmann-like equation for the one-particle phase space distribution $f(x,v,t)$. There are hard-sphere collisions between the particles as well as collisions with randomly fixed scatterers. As a result, this evolution does not conserve momentum but only mass and energy. We prove that the diffusively rescaled $f^\varepsilon(x,v,t)=f(\varepsilon^{-1}x,v,\varepsilon^{-2}t)$, as $\varepsilon\to 0$ tends to a Maxwellian $M_{ρ, 0, T}=\fracρ{(2πT)^{3/2}}\exp[{-\frac{|v|^2}{2T}}]$, where $ρ$ and $T$ are solutions of coupled diffusion equations and estimate the error in $L^2_{x,v}$.

math-ph↗

arXiv2019

Solution of the time dependent Schrödinger equation leading to Fowler-Nordheim field emission

We solve the time-dependent Schrödinger equation describing the emission of electrons from a metal surface by an external electric field $E$, turned on at $t=0$. Starting with a wave function $ψ(x,0)$, representing a generalized eigenfunction when $E=0$, we find $ψ(x,t)$ and show that it approaches, as $t\to\infty$, the Fowler-Nordheim tunneling wavefunction $ψ_E$. The deviation of $ψ$ from $ψ_E$ decays asymptotically as a power law $t^{-\frac32}$. The time scales involved for typical metals and fields of several V/nm are of the order of femtoseconds.

math-ph↗

arXiv2018

Transport properties of the classical Toda chain: effect of a pinning potential

We consider energy transport in the classical Toda chain in the presence of an additional pinning potential. The pinning potential is expected to destroy the integrability of the system and an interesting question is to see the signatures of this breaking of integrability on energy transport. We investigate this by a study of the non-equilibrium steady state of the system connected to heat baths as well as the study of equilibrium correlations. Typical signatures of integrable systems are a size-independent energy current, a flat bulk temperature profile and ballistic scaling of equilibrium dynamical correlations, these results being valid in the thermodynamic limit. We find that, as expected, these properties change drastically on introducing the pinning potential in the Toda model. In particular, we find that the effect of a harmonic pinning potential is drastically smaller at low temperatures, compared to a quartic pinning potential. We explain this by noting that at low temperatures the Toda potential can be approximated by a harmonic inter-particle potential for which the addition of harmonic pinning does not destroy integrability.

cond-mat.stat-mech↗

arXiv2018

High-fugacity expansion, Lee-Yang zeros and order-disorder transitions in hard-core lattice systems

We establish existence of order-disorder phase transitions for a class of "non-sliding" hard-core lattice particle systems on a lattice in two or more dimensions. All particles have the same shape and can be made to cover the lattice perfectly in a finite number of ways. We also show that the pressure and correlation functions have a convergent expansion in powers of the inverse of the fugacity. This implies that the Lee-Yang zeros lie in an annulus with finite positive radii.

math-ph↗

arXiv2018

On the nonequilibrium entropy of large and small systems

Thermodynamics makes definite predictions about the thermal behavior of macroscopic systems in and out of equilibrium. Statistical mechanics aims to derive this behavior from the dynamics and statistics of the atoms and molecules making up these systems. A key element in this derivation is the large number of microscopic degrees of freedom of macroscopic systems. Therefore, the extension of thermodynamic concepts, such as entropy, to small (nano) systems raises many questions. Here we shall reexamine various definitions of entropy for nonequilibrium systems, large and small. These include thermodynamic (hydrodynamic), Boltzmann, and Gibbs-Shannon entropies. We shall argue that, despite its common use, the last is not an appropriate physical entropy for such systems, either isolated or in contact with thermal reservoirs: physical entropies should depend on the microstate of the system, not on a subjective probability distribution. To square this point of view with experimental results of Bechhoefer we shall argue that the Gibbs-Shannon entropy of a nano particle in a thermal fluid should be interpreted as the Boltzmann entropy of a dilute gas of Brownian particles in the fluid.

cond-mat.stat-mech↗

arXiv2018

Diffusion equations from kinetic models with non-conserved momentum

We derive diffusive macroscopic equations for the particle and energy density of a system whose time evolution is described by a kinetic equation for the one particle position and velocity function f(r,v,t) that consists of a part that conserves energy and momentum such as the Boltzmann equation and an external randomization of the particle velocity directions that breaks the momentum conservation. Rescaling space and time by epsilon and epsilon square respectively and carrying out a Hilbert expansion in epsilon around a local equilibrium Maxwellian yields coupled diffusion equations with specified Onsager coefficients for the particle and energy density. Our analysis includes a system of hard disks at intermediate densities by using the Enskog equation for the collision kernel.

cond-mat.stat-mech↗

arXiv2018

Ballistic transport in the classical Toda chain with harmonic pinning

We investigate, via numerical simulation, heat transport in the nonequilibrium stationary state (NESS) of the 1D classical Toda chain with an additional pinning potential, which destroys momentum conservation. The NESS is produced by coupling the system, via Langevin dynamics, to two reservoirs at different temperatures. To our surprise, we find that when the pinning is harmonic, the transport is ballistic. We also find that on a periodic ring with nonequilibrium initial conditions and no reservoirs, the energy current oscillates without decay. Lastly, Poincaré sections of the 3-body case indicate that for all tested initial conditions, the dynamics occur on a 3-dimensional manifold. These observations suggest that the $N$-body Toda chain with harmonic pinning may be integrable. Alternatively, and more likely, this would be an example of a nonintegrable system without momentum conservation for which the heat flux is ballistic - contrary to all current expectations.

math-ph↗

arXiv2017

Ionization by an Oscillating Field: Resonances and Photons

We describe new exact results for a model of ionization of a bound state, induced by an oscillating potential. In particular we have obtained exact expressions, in the form of readily computable rapidly convergent sums, for the energy distribution of the emitted particles as a function of time, frequency and strength of the oscillating potential. Going beyond perturbation theory, these show resonances in the energy distribution which look like single or multi-photon absorption, similar to those observed in laser induced electron emission from solids or atoms. This is particularly so when the strength of the oscillating potential is small compared to the binding energy but is still visible for large fields, and even for time-periods of a few oscillations. We have also obtained the space-time structure of the wave function. Our model exhibits a form of stabilization; the ionization probability is not monotone in the strength of the oscillating potential.

math-ph↗

arXiv2017

Crystalline ordering and large fugacity expansion for hard core lattice particles

Using an extension of Pirogov-Sinai theory we prove phase transitions, corresponding to sublattice orderings, for a general class of hard core lattice particle systems with a finite number of close packed configurations. These include many cases for which such transitions have been proven. The proof also shows that, for these systems, the Gaunt-Fisher expansion of the pressure in powers of the inverse fugacity (aside from an explicit logarithmic term) has a nonzero radius of convergence.

cond-mat.stat-mech↗

arXiv2017

Nonperturbative time dependent solution of a simple ionization model

We present a non-perturbative solution of the Schrödinger equation $iψ_t(t,x)=-ψ_{xx}(t,x)-2(1 +α\sinωt) δ(x)ψ(t,x)$, written in units in which $\hbar=2m=1$, describing the ionization of a model atom by a parametric oscillating potential. This model has been studied extensively by many authors, including us. It has surprisingly many features in common with those observed in the ionization of real atoms and emission by solids, subjected to microwave or laser radiation. Here we use new mathematical methods to go beyond previous investigations and to provide a complete and rigorous analysis of this system. We obtain the Borel-resummed transseries (multi-instanton expansion) valid for all values of $α,ω,t$ for the wave function, ionization probability, and energy distribution of the emitted electrons, the latter not studied previously for this model. We show that for large $t$ and small $α$ the energy distribution has sharp peaks at energies which are multiples of $ω$, corresponding to photon capture. We obtain small $α$ expansions that converge for all $t$, unlike those of standard perturbation theory. We expect that our analysis will serve as a basis for treating more realistic systems revealing a form of universality in different emission processes.

math-ph↗

arXiv2017

Generalized stealthy hyperuniform processes : maximal rigidity and the bounded holes conjecture

We study translation invariant stochastic processes on $\mathbb{R}^d$ or $\mathbb{Z}^d$ whose diffraction spectrum or structure function $S(k)$, i.e. the Fourier transform of the truncated total pair correlation function, vanishes on an open set $U$ in the wave space. A key family of such processes are stealthy hyperuniform point processes, for which the origin $k=0$ is in $U$; these are of much current physical interest. We show that all such processes exhibit the following remarkable maximal rigidity : namely, the configuration outside a bounded region determines, with probability 1, the exact value (or the exact locations of the points) of the process inside the region. In particular, such processes are completely determined by their tail. In the 1D discrete setting (i.e. $\mathbb{Z}$-valued processes on $\mathbb{Z}$), this can also be seen as a consequence of a recent theorem of Borichev, Sodin and Weiss; in higher dimensions or in the continuum, such a phenomenon seems novel. For stealthy hyperuniform point processes, we prove the Zhang-Stillinger-Torquato conjecture that such processes have bounded holes (empty regions), with a universal bound that depends inversely on the size of $U$.

math.PR↗

arXiv2017

Time evolution of the Luttinger model with nonuniform temperature profile

We study the time evolution of a one-dimensional interacting fermion system described by the Luttinger model starting from a nonequilibrium state defined by a smooth temperature profile $T(x)$. As a specific example we consider the case when $T(x)$ is equal to $T_L$ ($T_R$) far to the left (right). Using a series expansion in $ε= 2(T_{R} - T_{L})/(T_{L}+T_{R})$, we compute the energy density, the heat current density, and the fermion two-point correlation function for all times $t \geq 0$. For local (delta-function) interactions, the first two are computed to all orders, giving simple exact expressions involving the Schwarzian derivative of the integral of $T(x)$. For nonlocal interactions, breaking scale invariance, we compute the nonequilibrium steady state (NESS) to all orders and the evolution to first order in $ε$. The heat current in the NESS is universal even when conformal invariance is broken by the interactions, and its dependence on $T_{L,R}$ agrees with numerical results for the $XXZ$ spin chain. Moreover, our analytical formulas predict peaks at short times in the transition region between different temperatures and show dispersion effects that, even if nonuniversal, are qualitatively similar to ones observed in numerical simulations for related models, such as spin chains and interacting lattice fermions.

cond-mat.stat-mech↗

arXiv2017

Approach to the steady state in kinetic models with thermal reservoirs at different temperatures

We continue the investigation of kinetic models of a system in contact via stochastic interactions with several spatially homogeneous thermal reservoirs at different temperatures. Considering models different from those investigated in earlier work, we explicitly compute the unique spatially uniform non-equilibrium steady state (NESS) and prove that it is approached exponentially fast from any uniform initial state. This leaves open the question of whether there exist NESS that are not spatially uniform. Making a further simplification of our models, we then prove non-existence of such NESS and exponential approach to the unique spatially uniform NESS (with a computably boundable rate). The method of proof relies on refined Doeblin estimates and other probabilisitic techniques, and is quite different form the analysis in earlier work that was based on contraction mapping methods.

math-ph↗

arXiv2017

Macroscopic and Microscopic Thermal Equilibrium

We study the nature of and approach to thermal equilibrium in isolated quantum systems. An individual isolated macroscopic quantum system in a pure or mixed state is regarded as being in thermal equilibrium if all macroscopic observables assume rather sharply the values obtained from thermodynamics. Of such a system (or state) we say that it is in macroscopic thermal equilibrium (MATE). A stronger requirement than MATE is that even microscopic observables (i.e., ones referring to a small subsystem) have a probability distribution in agreement with that obtained from the micro-canonical, or equivalently the canonical, ensemble for the whole system. Of such a system we say that it is in microscopic thermal equilibrium (MITE). The distinction between MITE and MATE is particularly relevant for systems with many-body localization (MBL) for which the energy eigenfuctions fail to be in MITE while necessarily most of them, but not all, are in MATE. However, if we consider superpositions of energy eigenfunctions (i.e., typical wave functions $ψ$) in an energy shell, then for generic macroscopic systems, including those with MBL, most $ψ$ are in both MATE and MITE. We explore here the properties of MATE and MITE and compare the two notions, thereby elaborating on ideas introduced in [Goldstein et al., Phys.Rev.Lett. 115: 100402 (2015)].

quant-ph↗

arXiv2016

Fluctuations, large deviations and rigidity in hyperuniform systems: a brief survey

We present a brief survey of fluctuations and large deviations of particle systems with subextensive growth of the variance. These are called hyperuniform (or superhomogeneous) systems. We then discuss the relation between hyperuniformity and rigidity. In particular we give sufficient conditions for rigidity of such systems in d=1,2.

math.PR↗

arXiv2016

A note on Lee-Yang zeros in the negative half-plane

We obtain lower bounds on the inverse compressibility of systems whose Lee-Yang zeros of the grand-canonical partition function lie in the left half of the complex fugacity plane. This includes in particular systems whose zeros lie on the negative real axis such as the monomer-dimer system on a lattice. We also study the virial expansion of the pressure in powers of the density for such systems. We find no direct connection between the positivity of the virial coefficients and the negativity of the L-Y zeros, and provide examples of either one or both properties holding. An explicit calculation of the partition function of the monomer-dimer system on 2 rows shows that there are at most a finite number of negative virial coefficients in this case.

cond-mat.stat-mech↗

arXiv2016

Steady states and universal conductance in a quenched Luttinger model

We obtain exact analytical results for the evolution of a 1+1-dimensional Luttinger model prepared in a domain wall initial state, i.e., a state with different densities on its left and right sides. Such an initial state is modeled as the ground state of a translation invariant Luttinger Hamiltonian $H_λ$ with short range non-local interaction and different chemical potentials to the left and right of the origin. The system evolves for time $t>0$ via a Hamiltonian $H_{λ'}$ which differs from $H_λ$ by the strength of the interaction. Asymptotically in time, as $t \to \infty$, after taking the thermodynamic limit, the system approaches a translation invariant steady state. This final steady state carries a current $I$ and has an effective chemical potential difference $μ_+ - μ_-$ between right- ($+$) and left- ($-$) moving fermions obtained from the two-point correlation function. Both $I$ and $μ_+ - μ_-$ depend on $λ$ and $λ'$. Only for the case $λ= λ' = 0$ does $μ_+ - μ_-$ equal the difference in the initial left and right chemical potentials. Nevertheless, the Landauer conductance for the final state, $G=I/(μ_+ - μ_-)$, has a universal value equal to the conductance quantum $e^2/h$ for the spinless case.

math-ph↗