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arXiv:2601.22247

Temperature as a Dynamically Maintained Steady State: Photonic Mechanisms, Maintenance Cost, and the Limits of the Infinite-Reservoir Idealization

David Vaknin

quant-phcond-mat.stat-mechSubmitted Jan 29, 2026• Announced Feb 2, 2026• Revised Feb 20, 2026 (v2)
v1 · Jan 29, 2026v2 · Feb 20, 2026v3 · Apr 12, 2026
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Abstract

Classical thermodynamics treats temperature as a state variable characterizing systems in equilibrium with idealized infinite reservoirs. We argue that this framing, while computationally exact, obscures an essential physical reality: any system at finite characteristic energy $E_c = k_B T$ continuously emits thermal radiation and cools unless energy input compensates these losses. What thermodynamics calls ``thermal equilibrium'' is, at the microscopic level, a dynamically sustained steady state maintained by continuous photon exchange. We derive that the average photon energy required to sustain a Planck distribution is $\langle hν\rangle = π^4 E_c/[30\,ζ(3)] \approx 2.701\,E_c$, quantifying the energetic throughput that any real system must sustain to maintain a given temperature. We resolve the apparent contradiction with the purely mechanical Maxwell velocity distribution: billiard-ball kinetics correctly describe the \emph{shape} of the distribution at a given $E_c$, but cannot account for how $E_c$ is established or maintained against radiative losses in any real system of charged particles. We further show that every finite thermal reservoir is itself maintained by photon exchange at a larger scale, organizing physical systems into a natural hierarchy from individual samples through cryostats, laboratories, and planetary surfaces to stellar interiors, with the classical infinite reservoir emerging as the large-capacity limit within that hierarchy rather than a fundamental physical entity. We also comment on the relation between thermodynamic entropy $S = k_B \ln W$ and the dimensionless entropy $\mathcal{S} = \ln W$, emphasizing that $k_B$ primarily fixes units (J/K) rather than introducing new statistical content. These results do not modify thermodynamics but provide its mechanistic interpretation in terms of quantum electrodynamics.

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Thermalization Without Equilibrium

Paul Cooney

Arc II — Observation Paper 1 Title Thermalization Without Equilibrium Zenodo DOI 10.5281/zenodo.18180820 Description This paper demonstrates that thermal behavior does not require equilibrium ensembles, microscopic ergodicity, or underlying statistical assumptions. Instead, thermalization is shown to arise from the progressive degradation of accessible correlations under finite observational capacity. The analysis reframes temperature, entropy production, and apparent equilibration as operational phenomena: when observers lose the ability to resolve fine-grained correlations, remaining observables converge toward thermal statistics regardless of the underlying dynamics. Thermal distributions therefore emerge as inference-fixed points under accessibility constraints, not as properties of microscopic phase space. By constructing thermalization without reference to microstates or equilibration timescales, the paper resolves long-standing conceptual tensions between thermodynamics and unitary dynamics. It clarifies why thermal behavior appears robust across systems with vastly different microscopic laws. This framework applies equally to classical, quantum, and gravitational settings, including horizon-induced thermality and detector responses in curved spacetime. Thermal behavior is shown to be a consequence of observational saturation rather than dynamical relaxation. Keywords thermalization; entropy production; irreversibility; operational thermodynamics; accessibility limits; non-equilibrium physics Notes Arc II — Observation (foundational) Thermal behavior derived without equilibrium assumptions Temperature as an inference artifact Applies to classical, quantum, and gravitational systems Conceptual bridge between entropy (Arc I) and measurement

•Jan 8, 2026•0 citations

Quantum and Classical Statistical Systems in Terms of Constraints

Francesco R. Ruggeri

Classical statistical mechanics/thermodynamics is sometimes portrayed as an approximate approach to describing the classical mechanical motion of very large numbers of particles. Equilibrium statistical mechanics/thermodynamics, however, is a special situation in which thermodynamic averages such as temperature, pressure etc do not change in time. We argue in this note that this is due to constraints imposed on such equilibrium scenarios namely time reversal reaction balance. It is this (approximate) constraint which yields the Maxwell-Boltzmann distribution which in turn defines average values of classical values e.g. pressure resulting from an average of bvv (b=constant) and temperature being related to average kinetic energy. Interestingly, average values such as average energy and pressure may yield a classical mechanical relationship for adiabatic processes, otherwise the more complicated dE=TdS - PdV applies to these averages. (To be more accurate one may use the constraints of microcanonical theory.) Quantum mechanics is also a statistical theory because a single particle with a fixed momentum is characterized by a physical wavelength region and a probability distribution exp(ipx). If interactions occur with potential regions a superposition of these distributions (i.e. a momentum superposition) occurs i.e. W(x)= Sum over p a(p)exp(ipx). This represents one constraint in quantum mechanics), but there is a second, namely that averages for a bound system match classical values. Thus, like classical statistical mechanics one may imagine a system characterized by a parameter say L (box length) which changes such that the particle stays in the same quantum state (i.e. an adiabatic change). Given that bound quantum mechanics is constrained to behave on average as a classical system it is not surprising that a dE=PdV which is essentially a Newton’s second law statement dE=Fdx and is not necessarily linked to a usual statistical system. If a quantum state has its parameter changed nonadiabatically it should be able to have it change into a superposition of new quantum states (new L) which is very different from the classical statistical mechanical scenario. Furthermore the ideal gas relationship PV=NRT and E=3/2NRT (3 degrees of freedom) or PV=(⅔)E matches a classical mechanical result because a particle with p may bounce from a wall yielding an impulse. In this note we try to examine some of these underlying constraints and argue that even dE=PdV for average E and P values (i.e. no entropy change) one must be aware of the specific constraints involved. In other words because dE=PdV for both adiabatic classical statistical mechanical systems and a quantum eigenstate, these do not necessarily represent the same statistical type of physics. We further argue that quantum mechanical bound states are constrained to have some average and rms values such as KE(x) and p rms behave as classical values and so these should not be taken to represent “intrinsic” quantum behaviour while average spatial density on the other hand does. (It is linked to the constraint of wavelength.)

•Oct 22, 2021•0 citations

Thermalization and Out-of-Equilibrium Dynamics in Open Quantum Many-Body Systems

Michael Buchhold

Thermalization, the evolution of an interacting many-body system towards a thermal Gibbs ensemble after initialization in an arbitrary non-equilibrium state, is currently a phenomenon of great interest, both in theory and experiment. As the time evolution of a quantum system is unitary, the proposed mechanism of thermalization in quantum many-body systems corresponds to the so-called eigenstate thermalization hypothesis (ETH) and the typicality of eigenstates. Although this formally solves the contradiction of thermalizing but unitary dynamics in a closed quantum many-body system, it does neither make any statement on the dynamical process of thermalization itself nor in which way the coupling of the system to an environment can hinder or modify the relaxation dynamics. In this thesis, we address both the question whether or not a quantum system driven away from equilibrium is able to relax to a thermal state, which fulfills detailed balance, and if one can identify universal behavior in the non-equilibrium relaxation dynamics. As a first realization of driven quantum systems out of equilibrium, we investigate a system of Ising spins, interacting with the quantized radiation field in an optical cavity. For multiple cavity modes, this system forms a highly entangled and frustrated state with infinite correlation times, known as a quantum spin glass. In the presence of drive and dissipation, introduced by coupling the intra-cavity radiation field to the photon vacuum outside the cavity via lossy mirrors, the quantum glass state is modified in a universal manner. For frequencies below the photon loss rate, the dissipation takes over and the system shows the universal behavior of a dissipative spin glass, with a characteristic spectral density $\\mathcal{A}(\\omega)\\sim\\sqrt{\\omega}$. On the other hand, for frequencies above the loss rate, the system retains the universal behavior of a zero temperature, quantum spin glass. Remarkably, at the glass transition, the two subsystems of spins and photons thermalize to a joint effective temperature, even in the presence of photon loss. This thermalization is a consequence of the strong spin-photon interactions, which favor detailed balance in the system and detain photons from escaping the cavity. In the thermalized system, the features of the spin glass are mirrored onto the photon degrees of freedom, leading to an emergent photon glass phase. Exploiting the inherent photon loss of the cavity, we make predictions of possible measurements on the escaping photons, which contain detailed information of the state inside the cavity and allow for a precise, non-destructive measurement of the glass state. As a further set of non-equilibrium systems, we consider one-dimensional quantum fluids driven out of equilibrium, whose universal low energy theory is formed by the so-called Luttinger Liquid description, which, due to its large degree of universality, is of intense theoretical and experimental interest. A set of recent experiments in research groups in Vienna, Innsbruck and Munich have probed the non-equilibrium time-evolution of one-dimensional quantum fluids for different experimental realizations and are pushing into a time regime, where thermalization is expected. From a theoretical point of view, one-dimensional quantum fluids are particular interesting, as Luttinger Liquids are integrable and therefore, due to an infinite number of constants of motion, do not thermalize. The leading order correction to the quadratic theory is irrelevant in the sense of the renormalization group and does therefore not modify static correlation functions, however, it breaks integrability and will therefore, even if irrelevant, induce a completely different non-equilibrium dynamics as the quadratic Luttinger theory alone. In this thesis, we derive for the first time a kinetic equation for interacting Luttinger Liquids, which describes the time evolution of the excitation densities for arbitrary initial states. The resonant character of the interaction makes a straightforward derivation of the kinetic equation, using Fermi\'s golden rule, impossible and we have to develop non-perturbative techniques in the Keldysh framework. We derive a closed expression for the time evolution of the excitation densities in terms of self-energies and vertex corrections. Close to equilibrium, the kinetic equation describes the exponential decay of excitations, with a decay rate $\\sigma^R=\\mbox{Im}\\Sigma^R$, determined by the self-energy at equilibrium. However, for long times $\\tau$, it also reveals the presence of dynamical slow modes, which are the consequence of exactly energy conserving dynamics and lead to an algebraic decay $\\sim\\tau^{-\\eta\\sub{D}}$ with $\\eta_D=0.58$. The presence of these dynamical slow modes is not contained in the equilibrium Matsubara formalism, while they emerge naturally in the non-equilibrium formalism developed in this thesis. In order to initialize a one-dimensional quantum fluid out of equilibrium, we consider an interaction quench in a model of interacting, dispersive fermions in Chap.~\\ref{sec:Quench}. In this scenario, the fermionic interaction is suddenly changed at time $t=0$, such that for $t>0$ the system is not in an eigenstate and therefore undergoes a non-trivial time evolution. For the quadratic theory, the stationary state in the limit $t\\rightarrow\\infty$ is a non-thermal, or prethermal, state, described by a generalized Gibbs ensemble (GGE). The GGE takes into account for the conservation of all integrals of motion, formed by the eigenmodes of the Hamiltonian. On the other hand, in the presence of non-linearities, the final state for $t\\rightarrow\\infty$ is a thermal state with a finite temperature $T>0$. . The spatio-temporal, dynamical thermalization process can be decomposed into three regimes: A prequench regime on the largest distances, which is determined by the initial state, a prethermal plateau for intermediate distances, which is determined by the metastable fixed point of the quadratic theory and a thermal region on the shortest distances. The latter spreads sub-ballistically $\\sim t^{\\alpha}$ in space with $0<\\alpha<1$ depending on the quench. Until complete thermalization (i.e. for times $t<\\infty$), the thermal region contains more energy than the prethermal and prequench region, which is expressed in a larger temperature $T_{t}>T_{\\infty}$, decreasing towards its final value $T_{\\infty}$. As the system has achieved local detailed balance in the thermalized region, energy transport to the non-thermal region can only be performed by the macroscopic dynamical slow modes and the decay of the temperature $T_{t}-T_{\\infty}\\sim t^{-\\eta\\sub{D}}$ again witnesses the presence of these slow modes. The very slow spreading of thermalization is consistent with recent experiments performed in Vienna, which observe a metastable, prethermal state after a quench and only observe the onset of thermalization on much larger time scales. As an immediate indication of thermalization, we determine the time evolution of the fermionic momentum distribution after a quench from non-interacting to interacting fermions. For this quench scenario, the step in the Fermi distribution at the Fermi momentum $k\\sub{F}$ decays to zero algebraically in the absence of a non-linearity but as a stretched exponential (the exponent being proportional to the non-linearity) in the presence of a finite non-linearity. This can serve as a proof for the presence or absence of the non-linearity even on time-scales for which thermalization can not yet be observed. Finally, we consider a bosonic quantum fluid, which is driven away from equilibrium by permanent heating. The origin of the heating is atomic spontaneous emission of laser photons, which are used to create a coherent lattice potential in optical lattice experiments. This process preserves the system\'s $U(1)$-invariance, i.e. conserves the global particle number, and the corresponding long-wavelength description is a heated, interacting Luttinger Liquid, for which phonon modes are continuously populated with a momentum dependent rate $\\partial_tn_q\\sim\\gamma |q|$. In the dynamics, we identify a quasi-thermal regime for large momenta, featuring an increasing time-dependent effective temperature. In this regime, due to fast phonon-phonon scattering, detailed balance has been achieved and is expressed by a time-local, increasing temperature. The thermal region emerges locally and spreads in space sub-ballistically according to $x_t\\sim t^{4/5}$. For larger distances, the system is described by an non-equilibrium phonon distribution $n_q\\sim |q|$, which leads to a new, non-equilibrium behavior of large distance observables. For instance, the phonon decay rate scales universally as $\\gamma_q\\sim |q|^{5/3}$, with a new non-equilibrium exponent $\\eta=5/3$, which differs from equilibrium. This new, universal behavior is guaranteed by the $U(1)$ invariant dynamics of the system and is insensitive to further subleading perturbations. The non-equilibrium long-distance behavior can be determined experimentally by measuring the static and dynamic structure factor, both of which clearly indicate the exponents for phonon decay, $\\eta=5/3$ and for the spreading of thermalization $\\eta_T=4/5$. Remarkably, even in the presence of this strong external drive, the interactions and their aim to achieve detailed balance are strong enough to establish a locally emerging and spatially spreading thermal region. The physical setups in this thesis do not only reveal interesting and new dynamical features in the out-of-equilibrium time evolution of interacting systems, but they also strongly underline the high degree of universality of thermalization for the classes of models studied here. May it be a system of coupled spins and photons, where the photons are pulled away from a thermal state by Markovian photon d

•Jun 29, 2015•1 citation

Thermal Operations from Informational Equilibrium

Seok Hyung Lie, Jeongrak Son, Paul Boës +2 more

Thermal operations are quantum channels that play a central role in deriving thermodynamic limitations in quantum systems. However, they were originally defined by implementation procedures rather than by fundamental principles. Alternative models of thermal processes have been proposed, but they obscure the resources required for implementation. Here, we identify the universal principle that separates these models. We show that thermal operations are uniquely characterized by a purely quantum information-theoretic property, which we call informational equilibrium. This formulation does not assume a particular Hamiltonian or even a notion of temperature: a channel is a thermal operation precisely when it leaves its environment invariant whenever the system is prepared in equilibrium. In this way, the usual energetic description of equilibrium (i.e., Gibbs states) emerges as a representation of an underlying informational constraint. Extending this framework, we note that enforcing environment invariance for all system inputs leads to catalytic channels, which idealize perfectly reusable heat baths. This informational equilibrium perspective culminates in a hierarchy for doubly-stochastic quantum channels, with strict separations between the classes. Our refined hierarchy sharpens the failure of the Birkhoff-von Neumann theorem in the quantum regime, and highlights the richer structural landscape of quantum channels compared to classical stochastic processes that cannot be captured by state-convertibility relations.

arXiv:2507.16637•Physical Review Letters•Jun 24, 2026•0 citations

Quantum thermodynamics of single particle systems

Md. Manirul Ali, Wei-Ming Huang, Wei-Min Zhang

Thermodynamics is built with the concept of equilibrium states. However, it is less clear how equilibrium thermodynamics emerges through the dynamics that follows the principle of quantum mechanics. In this paper, we develop a theory of quantum thermodynamics that is applicable for arbitrary small systems, even for single particle systems coupled with a reservoir. We generalize the concept of temperature beyond equilibrium that depends on the detailed dynamics of quantum states. We apply the theory to a cavity system and a two-level system interacting with a reservoir, respectively. The results unravels (1) the emergence of thermodynamics naturally from the exact quantum dynamics in the weak system-reservoir coupling regime without introducing the hypothesis of equilibrium between the system and the reservoir from the beginning; (2) the emergence of thermodynamics in the intermediate system-reservoir coupling regime where the Born-Markovian approximation is broken down; (3) the breakdown of thermodynamics due to the long-time non-Markovian memory effect arisen from the occurrence of localized bound states; (4) the existence of dynamical quantum phase transition characterized by inflationary dynamics associated with negative dynamical temperature. The corresponding dynamical criticality provides a border separating classical and quantum worlds. The inflationary dynamics may also relate to the origin of big bang and universe inflation. And the third law of thermodynamics, allocated in the deep quantum realm, is naturally proved.

arXiv:1803.04658•Scientific Reports•Aug 11, 2020•37 citations

The Thermodynamic Cost of Computing with Heat

M. W. AlMasri

Autonomous quantum thermal machines have recently been proposed as physics-based computing substrates where logical inputs and outputs are encoded in temperature gradients. While such ``thermodynamic neurons'' exhibit a clear trade-off between computational fidelity and heat dissipation, the fundamental information-theoretic limits of temperature-encoded computation remain uncharacterized. Here, we derive rigorous bounds linking average error probability, channel capacity, and entropy production for finite-capacity thermal reservoirs operating far from equilibrium. We prove that the minimal dissipation required to achieve a target average error probability $\langle ξ\rangle$ diverges as $\langle ξ\rangle$ approaches a fundamental minimum error floor $\epsMin$ imposed by finite-reservoir thermal fluctuations. We further establish a thermodynamic channel capacity that saturates at high dissipation, and quantify the minimal dissipation required for cascaded networks to maintain target fidelity. We demonstrate that the required dissipation grows linearly $\mathcal{O}(L)$ with network depth for shallow networks, but diverges as the network depth $L$ approaches a fundamental maximum limit $L_{\max} = \epsNet/\epsMin$ imposed by the minimum error floor. Furthermore, under strong noise amplification conditions, the required dissipation can grow up to $\mathcal{O}(L^3)$. Our framework bridges stochastic thermodynamics, finite-time information theory, and autonomous computation, providing rigorous design principles for energy-efficient analog thermodynamic hardware.

quant-pharXiv:2608.10027•Aug 11, 2026
Classical thermodynamics treats temperature as a state variable characterizing systems in equilibrium with idealized infinite reservoirs. We argue that this framing, while computationally exact, obscures an essential physical reality: any system at finite characteristic energy Ec=kBTE_c = k_B TEc​=kB​T continuously emits thermal radiation and cools unless energy input compensates these losses. What thermodynamics calls ``thermal equilibrium'' is, at the microscopic level, a dynamically sustained steady state maintained by continuous photon exchange.
We derive that the average photon energy required to sustain a Planck distribution is ⟨hν⟩=π4Ec/[30 ζ(3)]≈2.701 Ec\langle hν\rangle = π^4 E_c/[30\,ζ(3)] \approx 2.701\,E_c⟨hν⟩=π4Ec​/[30ζ(3)]≈2.701Ec​, quantifying the energetic throughput that any real system must sustain to maintain a given temperature. We resolve the apparent contradiction with the purely mechanical Maxwell velocity distribution: billiard-ball kinetics correctly describe the \emph{shape} of the distribution at a given EcE_cEc​, but cannot account for how EcE_cEc​ is established or maintained against radiative losses in any real system of charged particles. We further show that every finite thermal reservoir is itself maintained by photon exchange at a larger scale, organizing physical systems into a natural hierarchy from individual samples through cryostats, laboratories, and planetary surfaces to stellar interiors, with the classical infinite reservoir emerging as the large-capacity limit within that hierarchy rather than a fundamental physical entity. We also comment on the relation between thermodynamic entropy S=kBln⁡WS = k_B \ln WS=kB​lnW and the dimensionless entropy S=ln⁡W\mathcal{S} = \ln WS=lnW, emphasizing that kBk_BkB​ primarily fixes units (J/K) rather than introducing new statistical content. These results do not modify thermodynamics but provide its mechanistic interpretation in terms of quantum electrodynamics.
Arc II — Observation Paper 1 Title Thermalization Without Equilibrium Zenodo DOI 10.5281/zenodo.18180820 Description This paper demonstrates that thermal behavior does not require equilibrium ensembles, microscopic ergodicity, or underlying statistical assumptions. Instead, thermalization is shown to arise from the progressive degradation of accessible correlations under finite observational capacity. The analysis reframes temperature, entropy production, and apparent equilibration as operational phenomena: when observers lose the ability to resolve fine-grained correlations, remaining observables converge toward thermal statistics regardless of the underlying dynamics. Thermal distributions therefore emerge as inference-fixed points under accessibility constraints, not as properties of microscopic phase space. By constructing thermalization without reference to microstates or equilibration timescales, the paper resolves long-standing conceptual tensions between thermodynamics and unitary dynamics. It clarifies why thermal behavior appears robust across systems with vastly different microscopic laws. This framework applies equally to classical, quantum, and gravitational settings, including horizon-induced thermality and detector responses in curved spacetime. Thermal behavior is shown to be a consequence of observational saturation rather than dynamical relaxation. Keywords thermalization; entropy production; irreversibility; operational thermodynamics; accessibility limits; non-equilibrium physics Notes Arc II — Observation (foundational) Thermal behavior derived without equilibrium assumptions Temperature as an inference artifact Applies to classical, quantum, and gravitational systems Conceptual bridge between entropy (Arc I) and measurement
Classical statistical mechanics/thermodynamics is sometimes portrayed as an approximate approach to describing the classical mechanical motion of very large numbers of particles. Equilibrium statistical mechanics/thermodynamics, however, is a special situation in which thermodynamic averages such as temperature, pressure etc do not change in time. We argue in this note that this is due to constraints imposed on such equilibrium scenarios namely time reversal reaction balance. It is this (approximate) constraint which yields the Maxwell-Boltzmann distribution which in turn defines average values of classical values e.g. pressure resulting from an average of bvv (b=constant) and temperature being related to average kinetic energy. Interestingly, average values such as average energy and pressure may yield a classical mechanical relationship for adiabatic processes, otherwise the more complicated dE=TdS - PdV applies to these averages. (To be more accurate one may use the constraints of microcanonical theory.) Quantum mechanics is also a statistical theory because a single particle with a fixed momentum is characterized by a physical wavelength region and a probability distribution exp(ipx). If interactions occur with potential regions a superposition of these distributions (i.e. a momentum superposition) occurs i.e. W(x)= Sum over p a(p)exp(ipx). This represents one constraint in quantum mechanics), but there is a second, namely that averages for a bound system match classical values. Thus, like classical statistical mechanics one may imagine a system characterized by a parameter say L (box length) which changes such that the particle stays in the same quantum state (i.e. an adiabatic change). Given that bound quantum mechanics is constrained to behave on average as a classical system it is not surprising that a dE=PdV which is essentially a Newton’s second law statement dE=Fdx and is not necessarily linked to a usual statistical system. If a quantum state has its parameter changed nonadiabatically it should be able to have it change into a superposition of new quantum states (new L) which is very different from the classical statistical mechanical scenario. Furthermore the ideal gas relationship PV=NRT and E=3/2NRT (3 degrees of freedom) or PV=(⅔)E matches a classical mechanical result because a particle with p may bounce from a wall yielding an impulse. In this note we try to examine some of these underlying constraints and argue that even dE=PdV for average E and P values (i.e. no entropy change) one must be aware of the specific constraints involved. In other words because dE=PdV for both adiabatic classical statistical mechanical systems and a quantum eigenstate, these do not necessarily represent the same statistical type of physics. We further argue that quantum mechanical bound states are constrained to have some average and rms values such as KE(x) and p rms behave as classical values and so these should not be taken to represent “intrinsic” quantum behaviour while average spatial density on the other hand does. (It is linked to the constraint of wavelength.)
Thermalization, the evolution of an interacting many-body system towards a thermal Gibbs ensemble after initialization in an arbitrary non-equilibrium state, is currently a phenomenon of great interest, both in theory and experiment. As the time evolution of a quantum system is unitary, the proposed mechanism of thermalization in quantum many-body systems corresponds to the so-called eigenstate thermalization hypothesis (ETH) and the typicality of eigenstates. Although this formally solves the contradiction of thermalizing but unitary dynamics in a closed quantum many-body system, it does neither make any statement on the dynamical process of thermalization itself nor in which way the coupling of the system to an environment can hinder or modify the relaxation dynamics. In this thesis, we address both the question whether or not a quantum system driven away from equilibrium is able to relax to a thermal state, which fulfills detailed balance, and if one can identify universal behavior in the non-equilibrium relaxation dynamics. As a first realization of driven quantum systems out of equilibrium, we investigate a system of Ising spins, interacting with the quantized radiation field in an optical cavity. For multiple cavity modes, this system forms a highly entangled and frustrated state with infinite correlation times, known as a quantum spin glass. In the presence of drive and dissipation, introduced by coupling the intra-cavity radiation field to the photon vacuum outside the cavity via lossy mirrors, the quantum glass state is modified in a universal manner. For frequencies below the photon loss rate, the dissipation takes over and the system shows the universal behavior of a dissipative spin glass, with a characteristic spectral density mathcalA(omega)simsqrtomega\\mathcal{A}(\\omega)\\sim\\sqrt{\\omega}mathcalA(omega)simsqrtomega. On the other hand, for frequencies above the loss rate, the system retains the universal behavior of a zero temperature, quantum spin glass. Remarkably, at the glass transition, the two subsystems of spins and photons thermalize to a joint effective temperature, even in the presence of photon loss. This thermalization is a consequence of the strong spin-photon interactions, which favor detailed balance in the system and detain photons from escaping the cavity. In the thermalized system, the features of the spin glass are mirrored onto the photon degrees of freedom, leading to an emergent photon glass phase. Exploiting the inherent photon loss of the cavity, we make predictions of possible measurements on the escaping photons, which contain detailed information of the state inside the cavity and allow for a precise, non-destructive measurement of the glass state. As a further set of non-equilibrium systems, we consider one-dimensional quantum fluids driven out of equilibrium, whose universal low energy theory is formed by the so-called Luttinger Liquid description, which, due to its large degree of universality, is of intense theoretical and experimental interest. A set of recent experiments in research groups in Vienna, Innsbruck and Munich have probed the non-equilibrium time-evolution of one-dimensional quantum fluids for different experimental realizations and are pushing into a time regime, where thermalization is expected. From a theoretical point of view, one-dimensional quantum fluids are particular interesting, as Luttinger Liquids are integrable and therefore, due to an infinite number of constants of motion, do not thermalize. The leading order correction to the quadratic theory is irrelevant in the sense of the renormalization group and does therefore not modify static correlation functions, however, it breaks integrability and will therefore, even if irrelevant, induce a completely different non-equilibrium dynamics as the quadratic Luttinger theory alone. In this thesis, we derive for the first time a kinetic equation for interacting Luttinger Liquids, which describes the time evolution of the excitation densities for arbitrary initial states. The resonant character of the interaction makes a straightforward derivation of the kinetic equation, using Fermi\'s golden rule, impossible and we have to develop non-perturbative techniques in the Keldysh framework. We derive a closed expression for the time evolution of the excitation densities in terms of self-energies and vertex corrections. Close to equilibrium, the kinetic equation describes the exponential decay of excitations, with a decay rate sigmaR=mboxImSigmaR\\sigma^R=\\mbox{Im}\\Sigma^RsigmaR=mboxImSigmaR, determined by the self-energy at equilibrium. However, for long times tau\\tautau, it also reveals the presence of dynamical slow modes, which are the consequence of exactly energy conserving dynamics and lead to an algebraic decay simtau−etasubD\\sim\\tau^{-\\eta\\sub{D}}simtau−etasubD with etaD=0.58\\eta_D=0.58etaD​=0.58. The presence of these dynamical slow modes is not contained in the equilibrium Matsubara formalism, while they emerge naturally in the non-equilibrium formalism developed in this thesis. In order to initialize a one-dimensional quantum fluid out of equilibrium, we consider an interaction quench in a model of interacting, dispersive fermions in Chap.~\\ref{sec:Quench}. In this scenario, the fermionic interaction is suddenly changed at time t=0t=0t=0, such that for t>0t>0t>0 the system is not in an eigenstate and therefore undergoes a non-trivial time evolution. For the quadratic theory, the stationary state in the limit trightarrowinftyt\\rightarrow\\inftytrightarrowinfty is a non-thermal, or prethermal, state, described by a generalized Gibbs ensemble (GGE). The GGE takes into account for the conservation of all integrals of motion, formed by the eigenmodes of the Hamiltonian. On the other hand, in the presence of non-linearities, the final state for trightarrowinftyt\\rightarrow\\inftytrightarrowinfty is a thermal state with a finite temperature T>0T>0T>0. . The spatio-temporal, dynamical thermalization process can be decomposed into three regimes: A prequench regime on the largest distances, which is determined by the initial state, a prethermal plateau for intermediate distances, which is determined by the metastable fixed point of the quadratic theory and a thermal region on the shortest distances. The latter spreads sub-ballistically simtalpha\\sim t^{\\alpha}simtalpha in space with 0<alpha<10<\\alpha<10<alpha<1 depending on the quench. Until complete thermalization (i.e. for times t<inftyt<\\inftyt<infty), the thermal region contains more energy than the prethermal and prequench region, which is expressed in a larger temperature Tt>TinftyT_{t}>T_{\\infty}Tt​>Tinfty​, decreasing towards its final value TinftyT_{\\infty}Tinfty​. As the system has achieved local detailed balance in the thermalized region, energy transport to the non-thermal region can only be performed by the macroscopic dynamical slow modes and the decay of the temperature Tt−Tinftysimt−etasubDT_{t}-T_{\\infty}\\sim t^{-\\eta\\sub{D}}Tt​−Tinfty​simt−etasubD again witnesses the presence of these slow modes. The very slow spreading of thermalization is consistent with recent experiments performed in Vienna, which observe a metastable, prethermal state after a quench and only observe the onset of thermalization on much larger time scales. As an immediate indication of thermalization, we determine the time evolution of the fermionic momentum distribution after a quench from non-interacting to interacting fermions. For this quench scenario, the step in the Fermi distribution at the Fermi momentum ksubFk\\sub{F}ksubF decays to zero algebraically in the absence of a non-linearity but as a stretched exponential (the exponent being proportional to the non-linearity) in the presence of a finite non-linearity. This can serve as a proof for the presence or absence of the non-linearity even on time-scales for which thermalization can not yet be observed. Finally, we consider a bosonic quantum fluid, which is driven away from equilibrium by permanent heating. The origin of the heating is atomic spontaneous emission of laser photons, which are used to create a coherent lattice potential in optical lattice experiments. This process preserves the system\'s U(1)U(1)U(1)-invariance, i.e. conserves the global particle number, and the corresponding long-wavelength description is a heated, interacting Luttinger Liquid, for which phonon modes are continuously populated with a momentum dependent rate partialtnqsimgamma∣q∣\\partial_tn_q\\sim\\gamma |q|partialt​nq​simgamma∣q∣. In the dynamics, we identify a quasi-thermal regime for large momenta, featuring an increasing time-dependent effective temperature. In this regime, due to fast phonon-phonon scattering, detailed balance has been achieved and is expressed by a time-local, increasing temperature. The thermal region emerges locally and spreads in space sub-ballistically according to xtsimt4/5x_t\\sim t^{4/5}xt​simt4/5. For larger distances, the system is described by an non-equilibrium phonon distribution nqsim∣q∣n_q\\sim |q|nq​sim∣q∣, which leads to a new, non-equilibrium behavior of large distance observables. For instance, the phonon decay rate scales universally as gammaqsim∣q∣5/3\\gamma_q\\sim |q|^{5/3}gammaq​sim∣q∣5/3, with a new non-equilibrium exponent eta=5/3\\eta=5/3eta=5/3, which differs from equilibrium. This new, universal behavior is guaranteed by the U(1)U(1)U(1) invariant dynamics of the system and is insensitive to further subleading perturbations. The non-equilibrium long-distance behavior can be determined experimentally by measuring the static and dynamic structure factor, both of which clearly indicate the exponents for phonon decay, eta=5/3\\eta=5/3eta=5/3 and for the spreading of thermalization etaT=4/5\\eta_T=4/5etaT​=4/5. Remarkably, even in the presence of this strong external drive, the interactions and their aim to achieve detailed balance are strong enough to establish a locally emerging and spatially spreading thermal region. The physical setups in this thesis do not only reveal interesting and new dynamical features in the out-of-equilibrium time evolution of interacting systems, but they also strongly underline the high degree of universality of thermalization for the classes of models studied here. May it be a system of coupled spins and photons, where the photons are pulled away from a thermal state by Markovian photon d
Thermal operations are quantum channels that play a central role in deriving thermodynamic limitations in quantum systems. However, they were originally defined by implementation procedures rather than by fundamental principles. Alternative models of thermal processes have been proposed, but they obscure the resources required for implementation. Here, we identify the universal principle that separates these models. We show that thermal operations are uniquely characterized by a purely quantum information-theoretic property, which we call informational equilibrium. This formulation does not assume a particular Hamiltonian or even a notion of temperature: a channel is a thermal operation precisely when it leaves its environment invariant whenever the system is prepared in equilibrium. In this way, the usual energetic description of equilibrium (i.e., Gibbs states) emerges as a representation of an underlying informational constraint. Extending this framework, we note that enforcing environment invariance for all system inputs leads to catalytic channels, which idealize perfectly reusable heat baths. This informational equilibrium perspective culminates in a hierarchy for doubly-stochastic quantum channels, with strict separations between the classes. Our refined hierarchy sharpens the failure of the Birkhoff-von Neumann theorem in the quantum regime, and highlights the richer structural landscape of quantum channels compared to classical stochastic processes that cannot be captured by state-convertibility relations.
Thermodynamics is built with the concept of equilibrium states. However, it is less clear how equilibrium thermodynamics emerges through the dynamics that follows the principle of quantum mechanics. In this paper, we develop a theory of quantum thermodynamics that is applicable for arbitrary small systems, even for single particle systems coupled with a reservoir. We generalize the concept of temperature beyond equilibrium that depends on the detailed dynamics of quantum states. We apply the theory to a cavity system and a two-level system interacting with a reservoir, respectively. The results unravels (1) the emergence of thermodynamics naturally from the exact quantum dynamics in the weak system-reservoir coupling regime without introducing the hypothesis of equilibrium between the system and the reservoir from the beginning; (2) the emergence of thermodynamics in the intermediate system-reservoir coupling regime where the Born-Markovian approximation is broken down; (3) the breakdown of thermodynamics due to the long-time non-Markovian memory effect arisen from the occurrence of localized bound states; (4) the existence of dynamical quantum phase transition characterized by inflationary dynamics associated with negative dynamical temperature. The corresponding dynamical criticality provides a border separating classical and quantum worlds. The inflationary dynamics may also relate to the origin of big bang and universe inflation. And the third law of thermodynamics, allocated in the deep quantum realm, is naturally proved.
Autonomous quantum thermal machines have recently been proposed as physics-based computing substrates where logical inputs and outputs are encoded in temperature gradients. While such ``thermodynamic neurons'' exhibit a clear trade-off between computational fidelity and heat dissipation, the fundamental information-theoretic limits of temperature-encoded computation remain uncharacterized. Here, we derive rigorous bounds linking average error probability, channel capacity, and entropy production for finite-capacity thermal reservoirs operating far from equilibrium. We prove that the minimal dissipation required to achieve a target average error probability ⟨ξ⟩\langle ξ\rangle⟨ξ⟩ diverges as ⟨ξ⟩\langle ξ\rangle⟨ξ⟩ approaches a fundamental minimum error floor $\epsMin$ imposed by finite-reservoir thermal fluctuations. We further establish a thermodynamic channel capacity that saturates at high dissipation, and quantify the minimal dissipation required for cascaded networks to maintain target fidelity. We demonstrate that the required dissipation grows linearly O(L)\mathcal{O}(L)O(L) with network depth for shallow networks, but diverges as the network depth LLL approaches a fundamental maximum limit $L_{\max} = \epsNet/\epsMin$ imposed by the minimum error floor. Furthermore, under strong noise amplification conditions, the required dissipation can grow up to O(L3)\mathcal{O}(L^3)O(L3). Our framework bridges stochastic thermodynamics, finite-time information theory, and autonomous computation, providing rigorous design principles for energy-efficient analog thermodynamic hardware.