Elucidating fundamental limitations inherent in physical systems is a central subject in physics. For important thermodynamic operations such as information erasure, cooling, and copying, resources like time and energetic cost must be expended to achieve the desired outcome within a predetermined error margin. In the context of cooling, the unattainability principle of the third law of thermodynamics asserts that infinite "resources" are needed to reach absolute zero. However, the precise identification of relevant resources and how they jointly constrain achievable error remains unclear within the frameworks of stochastic and quantum thermodynamics. In this work, we introduce the concept of separated states, which consist of fully unoccupied and occupied states, and formulate the corresponding thermokinetic cost and error, thereby establishing a unifying framework for a broad class of thermodynamic operations. We then uncover a three-way trade-off relation between time, cost, and error for thermodynamic operations aimed at creating separated states, simply expressed as $τ\mathcal{C}\varepsilon_τ\ge 1-η$. This fundamental relation is applicable to diverse thermodynamic operations, including information erasure, cooling, and copying. It provides a profound quantification of the unattainability principle in the third law of thermodynamics in a general form. Building upon this relation, we explore the quantitative limitations governing cooling operations, the preparation of separated states, and a no-go theorem for exact classical copying. Furthermore, we extend these findings to the quantum regime, encompassing both Markovian and non-Markovian dynamics. Specifically, within Lindblad dynamics, we derive a similar three-way trade-off relation that quantifies the cost of achieving a pure state with a given error.
The quantum thermodynamic property of the fractional damping system is investigated extensively. A fractional power-law decaying entropy function is revealed which presents another evidence for the validity of the third law of thermodynamics in the quantum dissipative region. Several non-trivial characters are excavated such as that the entropy varies from a non-linear diverging function to a semi-linear decaying function of the fractional exponent as the temperature tends to absolute zero.
The first in a long series of papers by John T. Lewis, G. W. Ford and the present author, considered the problem of the most general coupling of a quantum particle to a linear passive heat bath, in the course of which they derived an exact formula for the free energy of an oscillator coupled to a heat bath in thermal equilibrium at temperature T. This formula, and its later extension to three dimensions to incorporate a magnetic field, has proved to be invaluable in analyzing problems in quantum thermodynamics. Here, we address the question raised in our title viz. Nernst's third law of thermo…
Force-current structure in Markovian open quantum systems and its applications: geometric housekeeping-excess decomposition and thermodynamic trade-off relations
The third law of thermodynamics in the form of the unattainability principle states that exact ground-state cooling requires infinite resources. Here we investigate the amount of non-equilibrium resources needed for approximate cooling. We consider as resource any system out of equilibrium, allowing for resources beyond the i.i.d. assumption and including the input of work as a particular case. We establish in full generality a sufficient and a necessary condition for cooling and show that for a vast class of non-equilibrium resources these two conditions coincide, providing a single necessary…
The quantum thermodynamic functions of a harmonic oscillator coupled to a heat bath through velocity-dependent coupling are obtained analytically. It is shown that both the free energy and the entropy decay fast with the temperature in relation to that of the usual coupling from. This implies that the velocity-dependent coupling helps to ensure the third law of thermodynamics.
We take the perspective of open quantum systems and examine from their nonequilibrium dynamics the conditions when the physical quantities, their relations and the laws of thermodynamics become well defined and viable for quantum many body systems. We first describe how an open system nonequilibrium dynamics (ONEq) approach is different from the closed combined system + environment in a global thermal state (CGTs) setup. Only after the open system equilibrates will it be amenable to conventional thermodynamics descriptions, thus quantum thermodynamics (QTD) comes at the end rather than assumed…
The consequences of the opening of a phason gap in incommensurate systems are studied on a simple model, the discrete frustrated $ϕ^4$-model. Analytical considerations and numerical results show that there is a very weak phase transition that can be characterized as third order.
Bento \textit{et al.} [Phys. Rev. E 91, 022105 (2015)] state that the Tsallis entropy violates the third law of thermodynamics for $q \leq 0$ and $0<q<1$. We show that their results are valid only for $q \geq 1$, since there is no distribution maximizing the Tsallis entropy for the intervals $q \leq 0$ and $0<q<1$ compatible with the system energy expression.
We have made a simple and natural modification of a recent quantum refrigerator model presented by Cleuren et al. in Phys. Rev, Lett.108, 120603 (2012). The original model consist of two metal leads acting as heat baths, and a set of quantum dots that allow for electron transport between the baths. It was shown to violate the dynamic third law of thermodynamics (the unattainability principle, which states that cooling to absolute zero in finite time is impossible), but by taking into consideration the finite energy level spacing in metals we restore the third law, while keeping all of the orig…
The Third Law of Thermodynamics constrains the phase diagram of{systems with a first-order quantum phase transition. For zero conjugate field, the coexistence curve has an infinite slope at T=0. If a tricritical point exists at T>0, then the associated tricritical wings are perpendicular to the T=0 plane, but not to the zero-field plane. These results are based on the third law and basic thermodynamics only, and are completely general. As an explicit example we consider the ferromagnetic quantum phase transition in clean metals, where a first-order quantum phase transition is commonly observed…
Bento \textit{et al.} [Phys. Rev. E 91, 022105 (2015)] recently stated that the Tsallis entropy violates the third law of thermodynamics for $0 < q <1$ in the sub-additive regime. We first show that the division between the regimes $q < 1$ and $q > 1$ is already inherent in the fundamental incomplete structure of the deformed logarithms and exponentials underlying the Tsallis entropy. Then, we provide the complete deformed functions and show that the Tsallis entropy conforms to the third law of thermodynamics for both super-additive $q < 1$ and sub-additive $q > 1$ regimes. Finally, we remark …
The thermodynamics expected of systems undergoing third order phase transition has been investigated by identifying the orders through the analytic continuation of the functional of the free energy, using Ehrenfest thermodynamic theory. We developed the Euler - Lagrange equations for the order parameter and the vector potential and solved them for the first time using well - known mathematical formulations.
Curado \textit{et al.} [Ann. Phys. \textbf{366} (2016) 22] have recently studied the axiomatic structure and the universality of a three-parameter trace-form entropy inspired by the group-theoretical structure. In this work, we study the group-theoretical entropy $S_{a,b,r}$ in the context of the third law of thermodynamics where the parameters $\left\lbrace a,b,r \right\rbrace $ are all independent. We show that this three-parameter entropy expression can simultaneously satisfy the third law of thermodynamics and the three Khinchin axioms, namely continuity, concavity and expansibility only w…
We comment that the model proposed in Phys. Rev. Lett. 108, 120603 (2012) violates the dynamical version of the third law of thermodynamics. We discuses the different formulations of the third law of thermodynamics and suggest a possible reason for the violation.
Connections between information theory and thermodynamics have proven to be very useful to establish bounding limits for physical processes. Ideas such as Landauer's erasure principle and information assisted work extraction have greatly contributed not only to enlarge our understanding about the fundamental limits imposed by nature, but also to enlighten the path for practical implementations of information processing devices. The intricate information-thermodynamics relation also entails a fundamental limit on parameter estimation, establishing a thermodynamic cost for information acquisitio…
The thermodynamic and kinetic uncertainty relations indicate trade-offs between the relative fluctuation of observables and thermodynamic quantities such as dissipation and dynamical activity. Although these relations have been well studied for classical systems, they remain largely unexplored in the quantum regime. In this paper, we investigate such trade-off relations for Markovian open quantum systems whose underlying dynamics are quantum jumps, such as thermal processes and quantum measurement processes. Specifically, we derive finite-time lower bounds on the relative fluctuation of both d…
We employ classical thermodynamics to gain information about absolute entropy, without recourse to statistical methods, quantum mechanics or the Third Law of thermodynamics. The Gibbs-Duhem equation yields various simple methods to determine the absolute entropy of a fluid. We also study the entropy of an ideal gas and the ionization of a plasma in thermal equilibrium. A single measurement of the degree of ionization can be used to determine an unknown constant in the entropy equation, and thus determine the absolute entropy of a gas. It follows from all these examples that the value of entrop…
The quantum thermodynamic behaviour of a charged oscillator in the presence of a magnetic field and coupled to a heat bath through different coupling schemes is obtained analytically. It is shown that finite dissipation substitutes the zero-coupling result of exponential decay of entropy by a power law behaviour at low temperature. For the coordinate-coordinates coupling scheme the low temperature explicit results for the case of Ohmic, exponentially correlated and more generalized heat bath models are derived. In all the above mentioned cases free energy and entropy vanish linearly with tempe…
The third law of thermodynamics has a controversial past and a number of formulations due to Planck, Einstein, and Nernst. It's most accepted version, the unattainability principle, states that "any thermodynamic process cannot reach the temperature of absolute zero by a finite number of steps and within a finite time." Although formulated in 1912, there has been no general proof of the principle, and the only evidence we have for it is that particular cooling methods become less efficient as the temperature decreases. Here we provide the first derivation of a general unattainability principle…
Thermodynamics, the branch of physics concerned with the description of macroscopic bodies, heat exchange and the conversion of different forms of energy is based on four laws: the zeroth law, which states that bodies in thermal contact reach a state of thermal equilibrium, the first law, which postulates the energy conservation and establishes the way in which different forms of energy transform into each-other, the second law, which states that the entropy increases or stays constant in time in any isolated system, and, finally, the third law (or Nernst postulate), which states that the entr…
The thermodynamic uncertainty relation sets the minimal bound of the cost-precision trade-off relation for dissipative processes. Examining the dynamics of an internally coupled system that is driven by a constant thermodynamic force, we however find that the trade-off relation of a sub-system is not constrained by the minimal bound of conventional uncertainty relation. We made our point explicit by using an exactly solvable model of interacting oscillators. As the number (N) of interacting oscillators increases, the uncertainty bound of individual oscillators is reduced to 2k_B T/N upon full …
The high accuracy exhibited by biological information transcription processes is due to kinetic proofreading, i.e., by a mechanism which reduces the error rate of the information-handling process by driving it out of equilibrium. We provide a consistent thermodynamic description of enzyme-assisted assembly processes involving competing substrates, in a Master Equation framework. We introduce and evaluate a measure of the efficiency based on rigorous non-equilibrium inequalities. The performance of several proofreading models are thus analyzed and the related time, dissipation and efficiency vs…
Finite-Time Thermodynamics of Battery Discharging: Power-Efficiency Trade-Off and Optimization
School of Physics and Astronomy, Beijing Normal University（北京师范大学物理学院）；Key Laboratory of Multiscale Spin Physics (Ministry of Education), Beijing Normal University（北京师范大学多尺度自旋物理教育部重点实验室）
Thermodynamic conditions ensure the stability of third-order extended heat conduction
HUN-REN Wigner Research Centre for Physics（匈牙利科学院维格纳物理研究中心）；Budapest University of Technology and Economics（布达佩斯科技大学）；Montavid Thermodynamic Research Group（蒙塔维德热力学研究组）；Eötvös Loránd University（厄特沃什·罗兰大学）
From a recent geometric generalization of Thermodynamic Uncertainty Relations (TURs) we derive novel upper bounds on the nonlinear response of an observable of an arbitrary system undergoing a change of probabilistic state. Various relaxations of these bounds allow to recover well known bounds such as (strengthenings of) Cramer-Rao's and Pinsker's inequalities. In particular we obtain a master inequality, named Symmetric Response Intensity Relation, which recovers several TURs as particular cases. We employ this set of bounds for three physical applications. First, we derive a trade-off betwee…
In this paper, we derive a number of inequalities which express power-efficiency trade-offs that hold generally for thermodynamic machines operating in non-equilibrium stationary states. One of these inequalities concerns the output power, which is bounded by a quadratic function of the thermodynamic efficiency multiplied by a factor. Different factors can be obtained according to the level of knowledge one has about the underlying dynamics of the machine, they can depend for instance on the covariance of the input flux, the dynamical activity, or the non-equilibrium conductance.
We consider open quantum systems consisting of a finite system of independent fermions with arbitrary Hamiltonian coupled to one or more equilibrium fermion reservoirs (which need not be in equilibrium with each other). A strong form of the third law of thermodynamics, $S(T) \rightarrow 0$ as $T\rightarrow 0$, is proven for fully open quantum systems in thermal equilibrium with their environment, defined as systems where all states are broadened due to environmental coupling. For generic open quantum systems, it is shown that $S(T)\rightarrow g\ln 2$ as $T\rightarrow 0$, where $g$ is the numbe…
Producing a large current typically requires large dissipation, as is the case in electric conduction, where Joule heating is proportional to the square of the current. Stochastic thermodynamics offers a framework to study nonequilibrium thermodynamics of small fluctuating systems, and quite recently, microscopic derivations and universal understanding of the trade-off relation between the current and dissipation have been put forward. Here we establish a universal framework clarifying how quantum coherence affects the trade-off between the current and dissipation: a proper use of coherence en…
In this work, low temperature thermodynamic behaviour in the context of dissipative diamagnetism with anomalous coupling is analyzed. We find that finite dissipation substitutes the zero-coupling result of exponential decay of entropy by a power law behaviour at low temperature. For Ohmic bath, entropy vanishes linearly with temperature, $T$, in conformity with Nernst's theorem. It is also shown that entropy decays faster in the presence of anomalous coupling than that of the usual coordinate-coordinate coupling. It is observed that velocity-velocity coupling is the most advantageous coupling …
Recently, it has been shown that there is a trade-off relation between thermodynamic cost and current fluctuations, referred to as the thermodynamic uncertainty relation (TUR). The TUR has been derived for various processes, such as discrete-time Markov jump processes and overdamped Langevin dynamics. For underdamped dynamics, it has recently been reported that some modification is necessary for application of the TUR. In this study, we present a more generalized TUR, applicable to a system driven by a velocity-dependent force in the context of underdamped Langevin dynamics, by extending the t…
The thermodynamic uncertainty relations provide a universal trade-offs between entropy dissipation rate and fluctuations in transport current. This relation has been mostly used to estimate a minimum entropy dissipation rate by experimentally measuring current fluctuations. Here we consider joint currents and show that such uncertainty relations cannot be simultaneously achievable for any two currents, which leads to a potential underestimation of the minimum entropy dissipation rate.
Although the third law of thermodynamics was established almost a century ago, it is not yet universally considered to be a fundamental law of physics. A major problem is that there are many materials having residual entropy. Amorphous materials and random alloy systems are well-known examples. A conventional view is that amorphous materials are not in thermodynamic equilibrium and must be exempted from the law. The recent development of material sciences has let to a variety of new materials. Some of them have ambiguous structures which do not fit the qualitative description of metastability.…
The Nernst formulation of the third law of ordinary thermodynamics (often referred to as the ``Nernst theorem'') asserts that the entropy, $S$, of a system must go to zero (or a ``universal constant'') as its temperature, $T$, goes to zero. This assertion is commonly considered to be a fundamental law of thermodynamics. As such, it seems to spoil the otherwise perfect analogy between the ordinary laws of thermodynamics and the laws of black hole mechanics, since rotating black holes in general relativity do not satisfy the analog of the ``Nernst theorem''. The main purpose of this paper is to …
A microscopic definition of the thermodynamic entropy in an isolated quantum system must satisfy (i) additivity, (ii) extensivity and (iii) the second law of thermodynamics. We show that the diagonal entropy, which is the Shannon entropy in the energy eigenbasis at each instant of time, meets the first two requirements and that the third requirement is satisfied if an arbitrary external operation is performed at typical times. In terms of the diagonal entropy, thermodynamic irreversibility follows from the facts that the Hamiltonian dynamics restricts quantum trajectories under unitary evoluti…
