Showing papers on "Thermal equilibrium published in 2023"

The kinetics of carbon pair formation in silicon prohibits reaching thermal equilibrium

Abstract: Thermal equilibrium is reached when the system assumes its lowest energy. This can be hindered by kinetic reasons; however, it is a general assumption that the ground state can be eventually reached. Here, we show that this is not always necessarily the case. Carbon pairs in silicon have at least three different configurations, one of them (B-configuration) is the G photoluminescence centre. Experiments revealed a bistable nature with the A-configuration. Electronic structure calculations predicted that the C-configuration is the real ground state; however, no experimental evidence was found for its existence. Our calculations show that the formation of the A- and B-configurations is strongly favoured over the most stable C-configuration which cannot be realized in a detectable amount before the pair dissociates. Our results demonstrate that automatized search for complex defects consisting of only the thermodynamically most stable configurations may overlook key candidates for quantum technology applications.

Nonequilibrium dynamics of axionlike particles: The quantum master equation

Abstract: We study the non-equilibrium dynamics of Axion-like particles (ALP) coupled to Standard Model degrees of freedom in thermal equilibrium. The Quantum Master Equation (QME) for the (ALP) reduced density matrix is derived to leading order in the coupling of the (ALP) to the thermal bath, but to \emph{all} orders of the bath couplings to degrees of freedom within or beyond the Standard Model other than the (ALP). The (QME) describes the damped oscillation dynamics of an initial misaligned (ALP) condensate, thermalization with the bath, decoherence and entropy production within a unifying framework. The (ALP) energy density $\mathcal{E}(t)$ features two components: a ``cold'' component from the misaligned condensate and a ``hot'' component from thermalization with the bath, with $\mathcal{E}(t)= \mathcal{E}_{c}\,e^{-\gamma(T)\,t}+\mathcal{E}_{h}(1-e^{-\gamma(T)\,t})$ thus providing a ``mixed dark matter'' scenario. Relaxation of the (ALP) condensate, thermalization, decoherence and entropy production occur on similar time scales. An explicit example with (ALP)-photon coupling, valid post recombination yields a relaxation rate $\gamma(T)$ with a substantial enhancement from thermal emission and absorption. A misaligned condensate is decaying at least since recombination and on the same time scale thermalizing with the cosmic microwave background (CMB). Possible consequences for birefringence of the (CMB) and (ALP) contribution to the effective number of ultrarelativistic species and galaxy formation are discussed.

Ultrafast Martensitic Phase Transition Driven by Intense Terahertz Pulses

Abstract: We report on an ultrafast nonequilibrium phase transition with a strikingly long-lived martensitic anomaly driven by above-threshold single-cycle terahertz pulses with a peak field of more than 1 MV/cm. A nonthermal, terahertz-induced depletion of low-frequency conductivity in Nb 3 Sn indicates increased gap splitting of high-energy Γ 12 bands by removal of their degeneracies, which induces the martensitic phase above their equilibrium transition temperature. In contrast, optical pumping leads to a Γ 12 gap thermal melting. Such light-induced nonequilibrium martensitic phase exhibits a substantially enhanced critical temperature up to ∼100 K, i.e., more than twice the equilibrium temperature, and can be stabilized beyond technologically relevant, nanosecond time scales. Together with first-principle simulations, we identify a compelling terahertz tuning mechanism of structural order via Γ 12 phonons to achieve the ultrafast phase transition to a metastable electronic state out of equilibrium at high temperatures far exceeding those for equilibrium states.

Model-independent bubble wall velocities in local thermal equilibrium

Abstract: Accurately determining bubble wall velocities in first-order phase transitions is of great importance for the prediction of gravitational wave signals and the matter-antimatter asymmetry. However, it is a challenging task which typically depends on the underlying particle physics model. Recently, it has been shown that assuming local thermal equilibrium can provide a good approximation when calculating the bubble wall velocity. In this paper, we provide a model-independent determination of bubble wall velocities in local thermal equilibrium. Our results show that, under the reasonable assumption that the sound speeds in the plasma are approximately uniform, the hydrodynamics can be fully characterized by four quantities: the phase strength αn , the ratio of the enthalpies in the broken and symmetric phases, Ψ n , and the sound speeds in both phases, cs and cb . We provide a code snippet that allows for a determination of the wall velocity and energy fraction in local thermal equilibrium in any model. In addition, we present a fit function for the wall velocity in the case cs = cb = 1/√(3).

Configurational temperature in active matter. II. Quantifying the deviation from thermal equilibrium

Abstract: This paper suggests using the configurational temperature $\Tc$ for quantifying how far an active-matter system is from thermal equilibrium. We measure this ``distance'' by the ratio of the systemic temperature $\Ts$ to $\Tc$, where $\Ts$ is the canonical-ensemble temperature for which the average potential energy is equal to that of the active-matter system. $\Tc$ is ``local'' in the sense that it is the average of a function, which only depends on how the potential energy varies in the vicinity of a given configuration; in contrast $\Ts$ is a global quantity. The quantity $\Ts/\Tc$ is straightforward to evaluate in a computer simulation; equilibrium simulations in conjunction with a single steady-state active-matter configuration are enough to determine $\Ts/\Tc$. We validate the suggestion that $\Ts/\Tc$ quantifies the deviation from thermal equilibrium by data for the radial distribution function of 3d Kob-Andersen and 2d Yukawa active-matter models with active Ornstein-Uhlenbeck and active Brownian Particle dynamics. Moreover, we show that $\Ts/\Tc$, structure, and dynamics of the homogeneous phase are all approximately invariant along the motility-induced phase separation (MIPS) boundary in the phase diagram of the 2d Yukawa model. The measure $\Ts/\Tc$ is not limited to active matter; it can be used for quantifying how far any system involving a potential-energy function, e.g., a driven Hamiltonian system, is from thermal equilibrium.

Evaluating the role of product gas composition in vibrational relaxation in pulsed microwave plasma-enhanced flames

Abstract: Hybrid fs/ps coherent anti-Stokes Raman scattering (CARS) is employed to investigate the vibrational temperature evolution of N2 in lean methane flames exposed to pulsed microwave irradiation. Vibrational temperature during and post microwave illumination by a 2 μs, 30 kW peak power, 3.05 GHz pulse is monitored in flames diluted with N2, N2 and CO2 , and N2 and Ar. Electric field strengths inside the microwave cavity are monitored directly using electric field probes. Temperature increases up to 140 K were observed in flames with additional Ar and CO2 dilution, whereas temperature increases by 80 K were observed in mixtures diluted with only N2 . The microwave energy deposition to excited states begins to thermalize over scales of 100 μs, however, equilibrium is not reached before excited combustion products convect out of the probe volume on the order of several 1 ms. Understanding the impact of varying bath gases on microwave interaction, magnitude of temperature rise and thermalization timescales is critical for the development and validation of new kinetic models for applications exhibiting significant degrees of thermal non-equilibrium, such as high-speed reentry flows and plasma-assisted combustion.

Universal phenomenology at critical exceptional points of nonequilibrium $O(N)$ models

Abstract: In thermal equilibrium the dynamics of phase transitions is largely controlled by fluctuation-dissipation relations: On the one hand, friction suppresses fluctuations, while on the other hand the thermal noise is proportional to friction constants. Out of equilibrium, this balance dissolves and one can have situations where friction vanishes due to antidamping in the presence of a finite noise level. We study a wide class of $O(N)$ field theories where this situation is realized at a phase transition, which we identify as a critical exceptional point. In the ordered phase, antidamping induces a continuous limit cycle rotation of the order parameter with an enhanced number of $2N-3$ Goldstone modes. Close to the critical exceptional point, however, fluctuations diverge so strongly due to the suppression of friction that in dimensions $d<4$ they universally either destroy a preexisting static order, or give rise to a fluctuation-induced first order transition. This is demonstrated within a non-perturbative approach based on Dyson-Schwinger equations for $N=2$, and a generalization for arbitrary $N$, which can be solved exactly in the long wavelength limit. We show that in order to realize this physics it is not necessary to drive a system far out of equilibrium: Using the peculiar protection of Goldstone modes, the transition from an $xy$ magnet to a ferrimagnet is governed by an exceptional critical point once weakly perturbed away from thermal equilibrium.

Thermodynamic bounds for diffusion in nonequilibrium systems with multiple timescales

Abstract: We derive a Thermodynamic Uncertainty Relation bounding the mean squared displacement of a Gaussian process with memory, driven out of equilibrium by unbalanced thermal baths and/or by external forces. Our bound is tighter with respect to previous results and also holds at finite time. We apply our findings to experimental and numerical data for a many-body interacting granular fluid, characterized by regimes of anomalous diffusion. In some cases, our relation can distinguish between equilibrium and non-equilibrium behavior, a non-trivial inference task, particularly for Gaussian processes.

Optimal time-entropy bounds and speed limits for Brownian thermal shortcuts

Abstract: By controlling in real-time the variance of the radiation pressure exerted on an optically trapped microsphere, we engineer temperature protocols that shortcut thermal relaxation when transferring the microsphere from one thermal equilibrium state to an other. We identify the entropic footprint of such accelerated transfers and derive optimal temperature protocols that either minimize the production of entropy for a given transfer duration or accelerate as much as possible the transfer for a given entropic cost. Optimizing the trade-off yields time-entropy bounds that put speed limits on thermalization schemes. We further show how optimization expands the possibilities for accelerating Brownian thermalization down to its fundamental limits. Our approach paves the way for the design of optimized, finite-time thermodynamic cycles at the mesoscale. It also offers a platform for investigating fundamental connections between information geometry and finite-time processes.

Energy measurements remain thermometrically optimal beyond weak coupling

Abstract: We develop a general perturbative theory of finite-coupling quantum thermometry up to second order in probe-sample interaction. By assumption the probe and sample are in thermal equilibrium, so the probe is described by the mean-force Gibbs state. We prove that the ultimate thermometric precision can be achieved--to second order in the coupling--solely by means of local energy measurements on the probe. Hence, seeking to extract temperature information from coherences or devising adaptive schemes confers no practical advantage in this regime. Additionally, we provide a closed-form expression for the quantum Fisher information, which captures the probe's sensitivity to temperature variations. Finally, we benchmark and illustrate the ease of use of our formulae with two simple examples. Our formalism is completely general and makes no assumptions about separation of dynamical timescales or the nature of either the probe or the sample. Therefore, by providing analytical insight into both the thermal sensitivity and the optimal measurement for achieving it, our results pave the way for quantum thermometry in setups where finite-coupling effects cannot be ignored.

Tolman-Ehrenfest’s criterion of thermal equilibrium extended to conformally static spacetimes

Abstract: With insight from examples and physical arguments, the Tolman-Ehrenfest criterion of thermal equilibrium for test fluids in static spacetimes is extended to local thermal equilibrium in conformally static geometries. The temperature of the conformally rescaled fluid scales with the inverse of the conformal factor, reproducing the evolution of the cosmic microwave background in Friedmann universes, the Hawking temperature of the Sultana-Dyer cosmological black hole, and a heuristic argument by Dicke.

Non-thermal particle acceleration and power-law tails via relaxation to universal Lynden-Bell equilibria

Abstract: Collisionless and weakly collisional plasmas often exhibit non-thermal quasi-equilibria. Among these quasi-equilibria, distributions with power-law tails are ubiquitous. It is shown that the statistical-mechanical approach originally suggested by Lynden-Bell (1967) can easily recover such power-law tails. Moreover, we show that, despite the apparent diversity of Lynden-Bell equilibria, a generic form of the equilibrium distribution at high energies is a `hard' power-law tail $\propto \varepsilon^{-2}$, where $\varepsilon$ is the particle energy. The shape of the `core' of the distribution, located at low energies, retains some dependence on the initial condition but it is the tail (or `halo') that contains most of the energy. Thus, a degree of universality exists in collisionless plasmas.

Model-independent bubble wall velocities in local thermal equilibrium

Abstract: Accurately determining bubble wall velocities in first-order phase transitions is of great importance for the prediction of gravitational wave signals and the matter-antimatter asymmetry. However, it is a challenging task which typically depends on the underlying particle physics model. Recently, it has been shown that assuming local thermal equilibrium can provide a good approximation when calculating the bubble wall velocity. In this paper, we provide a model-independent determination of bubble wall velocities in local thermal equilibrium. Our results show that, under the reasonable assumption that the sound speeds in the plasma are approximately uniform, the hydrodynamics can be fully characterized by four quantities: the phase strength $\alpha_n$, the ratio of the enthalpies in the broken and symmetric phases, $\Psi_n$, and the sound speeds in both phases, $c_s$ and $c_b$. We provide a code snippet that allows for a determination of the wall velocity and energy fraction in local thermal equilibrium in any model. In addition, we present a fit function for the wall velocity in the case $c_s = c_b = 1/\sqrt 3$.

Departures from equilibrium in the cathode region of an argon transferred arc at atmospheric pressure

Abstract: Several kinds of experimental diagnostics of the plasma in the cathode region of an argon transferred arc at atmospheric pressure have been performed. Previous studies based on emission spectroscopy in the stationary arc have shown that the plasma is not in equilibrium in this region: 1) the maximum intensity of ArI lines does not remain constant; 2) near the cathode, the values of the electron number density obtained by two methods are not consistent with each other; 3) the ratio of the excited level populations do not correspond to that of an equilibrium composition.
First, we present measurements of a stationary arc, at low current. The results are similar to those obtained at higher currents, showing that the departures from local thermodynamic equilibrium (LTE) are not explained by the classical criteria and must be associated with the convection of cold gas in the cathode region. Second, measurements have been performed during the arc extinction. They show that the thermal non-equilibrium (difference between the electron temperature and the gas temperature) is not detectable. Finally an analysis based on relaxation times of several mechanisms is proposed; some departures from equilibrium could be explained by an excitation non-equilibrium.

Generalization of Kirchhoff's Law of Thermal Radiation: The Inherent Relations Between Quantum Efficiency and Emissivity

Abstract: Planck's law of thermal radiation depends on the temperature, $T$, and the emissivity, $\epsilon$, of a body, where emissivity is the coupling of heat to radiation that depends on both phonon-electron nonradiative interactions and electron-photon radiative interactions. Another property of a body is absorptivity, $\alpha$, which only depends on the electron-photon radiative interactions. At thermodynamic equilibrium, nonradiative interactions are balanced, resulting in Kirchhoff's law of thermal radiation that equals these two properties, i.e., $\epsilon = \alpha$. For non-equilibrium, quantum efficiency ($QE$) describes the statistics of photon emission, which like emissivity depends on both radiative and nonradiative interactions. Past generalized Planck's equation extends Kirchhoff's law out of equilibrium by scaling the emissivity with the pump-dependent chemical-potential $\mu$, obscuring the relations between the body properties. Here we theoretically and experimentally demonstrate a prime equation relating these properties in the form of $\epsilon = \alpha(1-QE)$, which is in agreement with a recent universal modal radiation law for all thermal emitters. At equilibrium, these relations are reduced to Kirchhoff's law. Our work lays out the fundamental evolution of non-thermal emission with temperature, which is critical for the development of lighting and energy devices.

Thermal-bath effects in quantum quenches within quantum critical regimes

Abstract: We address the out-of-equilibrium dynamics arising from quantum-quench (QQ) protocols (instantaneous changes of the Hamiltonian parameters) in many-body systems within their quantum critical regime and in contact with thermal baths, homogeneously coupled to the systems. We consider two classes of QQ protocols. One of them uses the thermal bath to prepare the initial Gibbs state; then, after quenching, the thermal bath is removed and the dynamics of the system is unitary. Wealso address a more complex QQ protocol where the thermal bath is not removed after quenching, thus the quantum evolution is also driven by the interaction with the bath, which may be described by appropriate master equations for the density matrix of the system, where a further relevant time scale, or inverse decay rate, characterizes the system-bath coupling. Under these QQ protocols, the critical system develops out-of-equilibrium scaling behaviors, which extend those forisolated critical systems, by introducing further scaling variables proportional to the temperature and the decay rate associated with the thermal baths. These out-of-equilibrium scaling behaviors are checked by analyzing QQ protocols within fermionic Kitaev wires, or equivalently quantum Ising chains, supplemented with a particular modelization of thermal bath that guarantees the asymptotic thermalization within the Lindblad master equation for the dynamics of open systems.

Observing Dynamical Phases of a Bardeen-Cooper-Schrieffer Superconductor in a Cavity QED Simulator

Abstract: In conventional Bardeen-Cooper-Schrieffer (BCS) superconductors, electrons with opposite momenta bind into Cooper pairs due to an attractive interaction mediated by phonons in the material. While superconductivity naturally emerges at thermal equilibrium, it can also emerge out of equilibrium when the system's parameters are abruptly changed. The resulting out-of-equilibrium phases are predicted to occur in real materials and ultracold fermionic atoms but have not yet been directly observed. This work realizes an alternate way to generate the proposed dynamical phases using cavity quantum electrodynamics (cavity QED). Our system encodes the presence or absence of a Cooper pair in a long-lived electronic transition in $^{88}$Sr atoms coupled to an optical cavity and represents interactions between electrons as photon-mediated interactions through the cavity. To fully explore the phase diagram, we manipulate the ratio between the single-particle dispersion and the interactions after a quench and perform real-time tracking of subsequent dynamics of the superconducting order parameter using non-destructive measurements. We observe regimes where the order parameter decays to zero ("phase I"), assumes a non-equilibrium steady-state value ("phase II"), or exhibits persistent oscillations ("phase III") in the form of a self-generated Floquet phase. The capability to emulate these dynamical phases in optical cavities without real Cooper pairs demonstrates that programmable simulators can overcome many challenges faced by traditional approaches. This opens up exciting prospects for quantum simulation, including the potential to engineer unconventional superconductors and to probe beyond mean-field effects like the spectral form factor, and for increasing coherence time for quantum sensing.

The Sub-Exponential Critical Slowing Down at Floquet Time Crystal Phase Transition

Abstract: Critical slowing down (CSD) has been a trademark of critical dynamics for equilibrium phase transitions of a many-body system, where the relaxation time for the system to reach thermal equilibrium or quantum ground state diverges with system size. The time crystal phase transition has attracted much attention in recent years for it provides a scenario of phase transition of quantum dynamics, unlike conventional equilibrium phase transitions. Here, we study critical dynamics near the Floquet time crystal phase transition. Its critical behavior is described by introducing a space-time coarse grained correlation function, whose relaxation time diverges at the critical point revealing the CSD. This is demonstrated by investigating the Floquet dynamics of one-dimensional disordered spin chain. Through finite-size scaling analysis, we show the relaxation time has a universal sub-exponential scaling near the critical point, in sharp contrast to the standard power-law behavior for CSD in equilibrium phase transitions. This prediction can be readily tested in present quantum simulation experiments.

Thermal stability of hairy black holes

Abstract: We discuss thermodynamical stability for hairy black hole spacetimes, viewed as defects in the thermodynamical parameter space, taking into account the backreaction of a secondary hair onto the spacetime geometry, which is modified nontrivially. We derive, in a model independent way, the conditions for the hairy black hole with the secondary hair to reach a stable thermal equilibrium with the heat bath. Specifically, if the scalar hair, induced by interactions of the matter fields with quadratic-curvature corrections, produces an inner horizon in the deformed geometry, a thermodynamically stable configuration will be reached with the black hole becoming extremal in its final stage. We also attempt to make some conjectures concerning the implications of this thermal stability for the existence of a minimum length in a quantum spacetime.

Two-temperature combined diffusion coefficients in argon-helium thermal plasmas

Abstract: From the derivation of two-temperature diffusion number fluxes of species proposed by Rat et al and that of ambipolar diffusion coefficients, combined diffusion coefficients introduced by Murphy at equilibrium are extended to non-equilibrium thermal plasmas, where the kinetic temperature of electrons T_e is different from that of heavy species T_h .
Two-temperature ordinary and thermal combined diffusion coefficients are calculated in an argon-helium plasma at atmospheric pressure up to 30,000 K. Plasma composition is obtained using a non-equilibrium constant method. The dependence on the electron temperature of combined diffusion coefficients is analyzed for different mole percentages of argon in the mixture and for different values of the non-equilibrium parameter θ = T_e /T_h.
Similar behaviors to those obtained at equilibrium are highlighted with a decrease in the combined diffusion coefficients, at fixed electron temperature, as θ increases.

Thermalization of Weakly Coupled non-Abelian Plasmas at Next-to-leading Order

Abstract: We employ the QCD kinetic theory, including next-to-leading(NLO) order corrections in coupling constant, to study the evolution of weakly coupled non-Abelian plasmas towards thermal equilibrium. For two characteristic far-from-equilibrium systems with either under- or over-occupied initial conditions, the NLO corrections remain well under control for a wide range of couplings, and the overall effect of NLO corrections is a reduction in the time required for thermalization.

Stable non-equilibrium Fulde-Ferrell-Larkin-Ovchinnikov state in a spin-imbalanced driven-dissipative Fermi gas loaded on a three-dimensional cubic optical lattice

Abstract: We theoretically investigate a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) type superfluid phase transition in a driven-dissipative two-component Fermi gas. The system is assumed to be in the non-equilibrium steady state, which is tuned by adjusting the chemical potential difference between two reservoirs that are coupled with the system. Including pairing fluctuations by extending the strong-coupling theory developed in the thermal-equilibrium state by Nozieres and Schmitt-Rink to this non-equilibrium case, we show that a non-equilibrium FFLO (NFFLO) phase transition can be realized without spin imbalance, under the conditions that (1) the two reservoirs imprint a two-edge structure on the momentum distribution of Fermi atoms, and (2) the system is loaded on a three-dimensional cubic optical lattice. While the two edges work like two Fermi surfaces with different sizes, the role of the optical lattice is to prevent the NFFLO long-range order from destruction by NFFLO pairing fluctuations. We also draw the non-equilibrium mean-field phase diagram in terms of the chemical potential difference between the two reservoirs, a fictitious magnetic field to tune the spin imbalance of the system, and the environmental temperature of the reservoirs, to clarify the relation between the NFFLO state and the ordinary thermal-equilibrium FFLO state discussed in spin-imbalanced Fermi gases.

Tolman-Ehrenfest's criterion of thermal equilibrium extended to conformally static spacetimes

Abstract: With insight from examples and physical arguments, the Tolman-Ehrenfest criterion of thermal equilibrium for test fluids in static spacetimes is extended to local thermal equilibrium in conformally static geometries. The temperature of the conformally rescaled fluid scales with the inverse of the conformal factor, reproducing the evolution of the cosmic microwave background in Friedmann universes, the Hawking temperature of the Sultana-Dyer cosmological black hole, and a heuristic argument by Dicke.

General theory for discrete symmetry-breaking equilibrium states

Abstract: Spontaneous symmetry-breaking in phase transitions occurs when the system Hamiltonian is symmetric under a certain transformation, but the equilibrium states observed in nature are not. Here, we prove that when a discrete symmetry is spontaneously broken in a quantum system, then the time evolution necessarily conserves two additional and non-commuting quantities, besides the one linked to the symmetry. This implies the existence of equilibrium states consisting in superpositions of macroscopic configurations. Then, we propose an experimental realization of such equilibrium states with the current state-of-the art in quantum technologies. Through numerical calculations, we show that they survive as very long-lived pre-thermal states, even very far away from the thermodynamic limit. Finally, we also show that a small symmetry-breaking perturbation in the Hamiltonian stabilizes the conservation of one of the two former quantities, implying that symmetry-breaking equilibrium states become stable even in small quantum systems.

Nonequilibrium dynamics of fluctuations in an ultracold atomic mixture

Abstract: The nonequilibrium spin-changing dynamics of an ultracold mixture of lithium and sodium atoms reveals the importance of spin fluctuations for key observables. The authors show that the experimental control of fluctuations can give access to the dynamics of a long-lived metastable state, an instability region with strong growth of fluctuations, and a regime with an early approach to thermal equilibrium.

Plasma in Equilibrium

Abstract: BeforePlasma in equilibrium going into the details of plasma physics, Chap. 2 introduces basic physical quantities associated to plasma, which is in thermal equilibrium. The basic physical quantities include distribution function, number density, temperature, the Coulomb collision, etc.

Approach to Equilibrium of Statistical Systems: Classical Particles and Quantum Fields Off-Equilibrium

Abstract: Non-equilibrium evolution at absolute temperature T and approach to equilibrium of statistical systems in long-time (t) approximations, using both hierarchies and functional integrals, are reviewed. A classical non-relativistic particle in one spatial dimension, subject to a potential and a heat bath (hb), is described by the non-equilibrium reversible Liouville distribution (W) and equation, with a suitable initial condition. The Boltzmann equilibrium distribution Weq generates orthogonal (Hermite) polynomials Hn in momenta. Suitable moments Wn of W (using the Hn’s) yield a non-equilibrium three-term hierarchy (different from the standard Bogoliubov–Born–Green–Kirkwood–Yvon one), solved through operator continued fractions. After a long-t approximation, the Wn’s yield irreversibly approach to equilibrium. The approach is extended (without hb) to: (i) a non-equilibrium system of N classical non-relativistic particles interacting through repulsive short range potentials and (ii) a classical ϕ4 field theory (without hb). The extension to one non-relativistic quantum particle (with hb) employs the non-equilibrium Wigner function (WQ): difficulties related to non-positivity of WQ are bypassed so as to formulate approximately approach to equilibrium. A non-equilibrium quantum anharmonic oscillator is analyzed differently, through functional integral methods. The latter allows an extension to relativistic quantum ϕ4 field theory (a meson gas off-equilibrium, without hb), facing ultraviolet divergences and renormalization. Genuine simplifications of quantum ϕ4 theory at high T and large distances and long t occur; then, through a new argument for the field-theoretic case, the theory can be approximated by a classical ϕ4 one, yielding an approach to equilibrium.