Relaxation of a Free-Molecular Gas to Equilibrium Caused by Interaction with Vessel Wall

A free-molecular gas contained in a static vessel with a uniform temperature is considered. The approach of the velocity distribution function of the gas molecules from a given initial distribution to the uniform equilibrium state at rest is investigated numerically under the diffuse reflection boundary condition. This relaxation is caused by the interaction of gas molecules with the vessel wall. It is shown that, for a spherical vessel, the velocity distribution function approaches the final uniform equilibrium distribution in such a way that their difference decreases in proportion to an inverse power of time. This is slower than the known result for a rarefied gas with molecular collisions.     

The Lifshitz Tail and Relaxation to Equilibrium in the One-Dimensional Disordered Ising Model

We study spectral properties of the generator of the Glauber dynamics for a 1D disordered stochastic Ising model with random bounded couplings. By an explicit representation for the upper branch of the generator we get an asymptotic formula for the integrated density of states of the generator near the upper edge of the spectrum. This asymptotic behavior appears to have the form of the Lifshitz tail, which is typical for random operators near fluctuation boundaries. As a consequence we find the asymptotics for the average over the disorder of the time-autocorrelation function to be $$\langle \langle \sigma _{\text{0}}^\omega (t),\sigma _0 (0)\rangle _{P(\omega ) = } {\text{exp\{ }} - gt - kt^{1/3} {\text{(1 + }}o(1){\text{)\} as }}t \to \infty $$ with constants g, k depending on the distribution of the random couplings.     

Relaxation to Equilibrium for Two Dimensional Disordered Ising Systems in the Griffiths Phase

We consider Glauber–type dynamics for two dimensional disordered magnets of Ising type. We prove that, if the disorder–averaged influence of the boundary condition is sufficiently small in the equilibrium system, then the corresponding Glauber dynamics is ergodic with probability one and the disorder–average C(t) of time–autocorrelation function satisfies (for large t). For the standard two dimensional dilute Ising ferromagnet with i.i.d. random nearest neighbor couplings taking the values 0 or J ₀>0, our results apply even if the active bonds percolate and J ₀ is larger than the critical value J _c of the corresponding pure Ising model. For the same model we also prove that in the whole Griffiths' phase the previous upper bound is optimal. This implies the existence of a dynamical phase transition which occurs when J crosses J _c . Received:     

Two-Time-Scale Relaxation Towards Thermal Equilibrium of the Enigmatic Piston

We investigate the evolution of a system composed of N non-interacting point particles of mass m in a container divided into two chambers by a movable adiabatic piston of mass Mm. Using a two-time-scale perturbation approach in terms of the small parameter =2m/(M+m), we show that the evolution towards thermal equilibrium proceeds in two stages. The first stage is a fast, deterministic, adiabatic relaxation towards mechanical equilibrium. The second stage, which takes place at times (M), is a slow fluctuation-driven, diathermic relaxation towards thermal equilibrium. A very simple equation is derived which shows that in the second stage, the position of the piston is given by X _M (t)= L[1/2–(t)] where the function is independent of M. Numerical simulations support the assumptions underlying our analytical derivations and illustrate the large mass range in which the picture holds.     

Trend to Equilibrium of a Degenerate Relativistic Gas

We examine the problem of the trend to equilibrium for a relativistic gas which may follow Fermi–Dirac, Bose–Einsten, classical Boltzmann statistics. We use the relativistic version of the quasiclassical Boltzmann equation for fermions and bosons, the Uehling–Uhlenbeck equation.     

Pulsating optical source appearing under continuous laser energy supply to a hypersonic air flow

The spatio-temporal gasdynamic structures of plasma formations produced in a hypersonic air flow by a focused high-power laser beam are analyzed numerically using the model of inviscid non-heat-conducting equilibrium radiating air. The reasons for the emergence of pulsating gasdynamic nonuniformity in the region between the bow shock wave and the intense laser radiation absorption zone are investigated.     

Symmetry of Pulsating Ratchets

Using an exact expression for the average velocity of inertialess motion of pulsating ratchets, a simple proof is given for the recently discovered hidden space-time symmetry of Cubero–Renzoni (D. Cubero, F. Renzoni, 2016). The conditions are revealed for the absence of the ratchet effect in systems with potential energies described by products of periodic functions of coordinate and time possessing the symmetry of the main types. In particular, it is shown that the ratchet effect is absent for the time dependence of the universal symmetry type (which combines three standard symmetries), and this restriction is removed when inertia is taken into account, unless the coordinate dependence of the potential energy is related to symmetric or antisymmetric functions.     

Relaxation in a fermi liquid

The initial value problem for fermions is solved by an explicit construction of the time evolution operator in terms of a spectral representation of the linearized Uehling-Uhlenbeck collision integral. Unless the scattering between the fermions is very anisotropic the decay of a perturbation follows a universal law with relaxation frequencies depending on the total cross section and the ratio of the free energy per particle to the thermal broadening of the Fermi surface. The theory is applied to the decay of quasi-particles and quadrupole deformations of the Fermi sphere. Also the thermalization of the energy of relative motion in a heavy ion collision is studied. For an investigation of the general properties of the linearized fermion Uehling-Uhlenbeck collision operator a Levinson theorem for one-dimensional problems is derived.     

Approach to Equilibrium in a Microscopic Model of Friction

We consider the time evolution of a disk under the action of a constant force and interacting with a free gas in the mean-field approximation. Letting V₀>0 be the initial velocity of the disk and V_∞>0 its equilibrium velocity, namely the one for which the external field is balanced by the friction force exerted by the background, we show that, if V_∞−V₀ is positive and sufficiently small, then the disk reaches V_∞ with the power law t^−(d+2), d=1,2,3 being the dimension of the physical space. The reason for this behavior is the long tail memory due to recollisions. Any Markovian approximation (or simply neglecting the recollisions) yields an exponential approach to equilibrium.     

Relaxation in a Duffing potential

The solution of the equation of motion for a particle in a Duffing potential, V(x) = α₁x²/2 + α₂x⁴/4 (α₁, α₂ > 0) for arbitrary anharmonicity strength is characterized by the presence of odd frequencies which implies that velocity and position autocorrelation functions of such an oscillator in a microcanonical ensemble are also characterized by odd frequencies. It is, however, non-trivial to determine whether such “discrete” frequencies also characterize the autocorrelation functions in a canonical ensemble as discussed recently by Fronzoni et al. (J. Stat. Phys. 41 (1985) 553). We recover and extend upon the results of Fronzoni et al. to show analytically, via Mori-Lee theory, that “essentially discrete” (i.e. well-defined peaks with finite but “small” width) temperature-dependent frequencies characterize the autocorrelation functions in a canonical ensemble.     

脉动参数对波纹通道内传热强化的影响

采用曲线坐标系下压力与速度耦合的SIMPLER算法,数值研究了波纹通道内脉动流动与换热情况,流动Re数的范围为5~500,Pr数为0.7.计算考察了脉动参数如脉动频率和振幅对通道内强化传热和压力损失的影响.研究结果表明,流动阻力特性呈周期性余弦规律变化,换热Nu数呈正弦规律变化;频率、振幅的增大,使得阻力脉动幅度增大.受入口脉动流的影响,通道内的旋涡发生周期性的脱落、增长和迁移,从而增强了流体之间的扰动和掺混,强化了传热;传热的强化效果随着振幅的增大而增强,但在特定入口脉动流下,相同振幅不同频率下的强化效果几乎一致.     

Pulsating and synthetic impinging jets

This is the second part of a survey summarizing authors’ research over a period of two decades on enhancing impinging jet heat and/or mass transfer by periodic unsteadiness of nozzle flow rate. The first part, Tesař and Trávníček (2004 b), identified the reasons why pulsation does not always improve the transfer rate: the pulses do not reach up to the wall. The authors nevertheless demonstrate a transfer rate improvement, but in flows with inherent instability found in annular impinging jet. The excitation there causes a topological metamorphosis (reversal of flow character). Also in the extreme case of the synthetic (zero time-mean flow) jets the authors demonstrated a substantial improvement with the annular nozzle. The new approaches presented in the paper offer increased performance of drying and heating/cooling systems, in particular in microdevices with otherwise low or absent levels of natural jet turbulence.     

Relaxation of a two-level system coupled to a Bose gas with Einstein spectrum

This note contains the exact diagonalisation of the Hamiltonian describing a two-level system coupled to a Bose gas with an Einstein spectrum (Sections II–IV). In Sections VI and VII we calculate the decay in time of different initial situations of the system.     

Relaxation of a dislocation in a nanocrystallite

Experimental studies of dislocations in nanoparticles are just beginning. The corresponding theoretical models are still lacking. In this context, the author analyzes relaxation of a dislocation in a nanoparticle. Mechanistically, this process is considered to occur primarily via dislocation drift induced by the stress-related image forces. Elementary dislocation displacements include the formation of a kink at one of the sides of the dislocation line, its diffusion along this line, and annihilation at the opposite side. For this mechanism, the dependence of the time of dislocation disappearance on the nanoparticle size has been identified.     

Pulsating instability of a Bose-Einstein condensate in an optical lattice

We find numerically that in the limit of weak atom-atom interactions a Bose-Einstein condensate in an optical lattice may develop a pulsating dynamical instability in which the atoms nearly periodically form a peak in the occupation numbers of the lattice sites, and then return to the unstable initial state. Multiple peaks behaving similarly are also found. Simple arguments show that the pulsating instability is a remnant of integrability, and give a handle on the relevant physical scales.     

Pulsating stars in general relativity

The equation governing radial pulsations of fully relativistic stars is derived and expressed in terms of quantities which are continuous even across density discontinuities which occur, e.g., in zero-temperature stellar models that undergo electron capture. When expressed in terms of these quantities, the pulsation equation can be integrated through density discontinuities without any special treatment of these points being necessary. Expressions for the adiabatic index and pulsation energy are derived in a simple way.Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under AFOSR Grant 70-1866.     

Thermal Relaxation of a QED Cavity

We study repeated interactions of the quantized electromagnetic field in a cavity with single two-levels atoms. Using the Markovian nature of the resulting quantum evolution we study its large time asymptotics. We show that, whenever the atoms are distributed according to the canonical ensemble at temperature T>0 and some generic non-degeneracy condition is satisfied, the cavity field relaxes towards some invariant state. Under some more stringent non-resonance condition, this invariant state is thermal equilibrium at some renormalized temperature T ^*. Our result is non-perturbative in the strength of the atom-field coupling. The relaxation process is slow (non-exponential) due to the presence of infinitely many metastable states of the cavity field.