Quantum Laws Redefined Reveal How Energy Loss Drives Systems Out of Balance

Scientists are increasingly focused on understanding thermodynamic behaviour in systems far from equilibrium, a challenge intensified when quantum coherence is present. Md Manirul Ali from the Department of Physics, Chennai Institute of Technology, and Po-Wen Chen from the National Atomic Research Institute, along with colleagues, have developed a novel first-principles framework addressing this issue by integrating coherence resource theory with established thermodynamic laws. Their research derives a previously unknown entropy balance relation which distinctly separates entropy flux from the loss of coherence, crucially identifying energy entropy as the relevant measure for nonequilibrium processes, rather than the commonly used von Neumann entropy. This work consistently defines key quantities like dynamical temperature and free energy, upholding both the first and second laws even in far-from-equilibrium scenarios, and offers a unified foundation for studying nonequilibrium dynamics and the fundamental role of quantum coherence.

They derive a previously unexplored entropy balance relation that explicitly separates entropy flux due to heat exchange from entropy production arising from the loss of quantum coherence.

This formulation identifies the appropriate thermodynamic entropy in nonequilibrium quantum processes as the energy entropy associated with energy measurements, demonstrating that the von Neumann entropy does not, in general, represent thermodynamic entropy away from equilibrium. Within this framework, dynamical temperature, free energy, work, and heat are consistently defined, and both the first and second laws are shown to hold far from equilibrium.
Applying the theory to an exactly solvable open quantum system, researchers reveal how equilibrium thermodynamics emerges dynamically in the weak-coupling limit. These results establish a unified and operational foundation for nonequilibrium quantum thermodynamics and clarify the fundamental thermodynamic role of quantum coherence.

Thermodynamic systems at the microscopic level, such as atoms, molecules, and photons, obey the laws of quantum mechanics. Thermodynamics and statistical mechanics are traditionally founded on the hypothesis of equilibrium. Understanding how equilibrium thermodynamics arises from underlying quantum dynamics, and how thermodynamic laws extend beyond equilibrium, remains a central challenge in modern physics.
However, a comprehensive theory of nonequilibrium quantum thermodynamics has not yet been established. Describing thermodynamic processes far from equilibrium requires a deep knowledge of the dynamics of systems interacting with their environment, for which the theory of open quantum systems provides a natural and rigorous framework.

Quantum thermodynamics seeks to formulate thermodynamic concepts such as entropy, heat, work, temperature, and free energy at the quantum scale and to analyse their behaviour in quantum engines and refrigerators. Extending these concepts consistently beyond equilibrium remains an open and fundamental problem.

At the heart of nonequilibrium quantum thermodynamics lies the definition of thermodynamic entropy S(t) and its relation to heat exchange. In many studies, nonequilibrium entropy is identified with the von Neumann entropy S(t), multiplied by the Boltzmann constant, and this identification is routinely employed in analyses of thermodynamic laws and quantum thermal device modelling.

While this identification is valid in thermal equilibrium, where the system is described by a Gibbs state ρ = e−βH Z, Z = tr{e−βH}, and the density matrix is diagonal in the energy eigenbasis, the situation changes fundamentally away from equilibrium. In equilibrium, the von Neumann entropy S = −tr{ρ ln ρ} coincides with the thermodynamic entropy S.

However, extending this equivalence to nonequilibrium quantum states is generally unjustified, particularly in the presence of quantum coherence. In fact, von Neumann entropy cannot, in general, represent thermodynamic entropy for coherent nonequilibrium states. Entropy plays a central role in both thermodynamics and information theory, reflecting a deep connection between thermodynamic processes and information.

Thermodynamic irreversibility in open quantum systems is quantitatively characterised by entropy production during nonequilibrium processes. When a quantum system interacts with a thermal reservoir, it may exchange energy, information, or particles. Researchers restrict attention to energy and information exchange in the absence of particle transfer.

Coupling the system to a thermal reservoir induces dissipation, reflected in the redistribution of energy populations, and decoherence, manifested by the decay of off-diagonal elements of the density matrix in the energy eigenbasis. These processes drive the system irreversibly toward equilibrium. The entropy production in such nonequilibrium dynamics has two distinct physical origins: one associated with heat exchange between the system and the reservoir, causing a rearrangement of the diagonal elements of the system density matrix, and another arising from information exchange due to the loss of quantum coherence.

Far from equilibrium, these two contributions compete dynamically, yet a unified and quantitative description of their interplay has remained elusive. Although recent studies have highlighted the role of quantum coherence in thermodynamic processes, a fully unified framework that incorporates coherence directly into the laws of nonequilibrium thermodynamics has remained incomplete.

In this work, scientists integrate quantum information resource theory of coherence and quantum thermodynamics through an unexplored entropy balance equation in nonequilibrium. The new entropy balance relation explicitly separates entropy flux due to heat exchange from entropy production arising from information exchange or coherence loss, thereby identifying the appropriate thermodynamic entropy governing nonequilibrium thermodynamics in the presence of coherence.

Within this framework, dynamical definitions of temperature, free energy, work, and heat emerge naturally, and both the first and second laws of thermodynamics are shown to hold throughout the nonequilibrium evolution. Using an exactly solvable open quantum system, they further demonstrate how equilibrium thermodynamics is dynamically recovered in the weak-coupling limit.

An open quantum system coupled to a thermal reservoir is described by the total Hamiltonian, Htot(t) = H(t) + HR + HI, where H(t) and HR are system and reservoir Hamiltonian, and HI describes the interaction between them. The total density matrix evolves under the Liouville von Neumann equation, iħ ρtot(t) = [Htot(t), ρtot(t)].

The reservoir is initially considered in a thermal equilibrium state, and the system is initially in an arbitrary state so that the initial total density matrix is a factorized state. The evolution of the open quantum system is entirely captured by its reduced density operator ρ(t), obtained by eliminating the environmental degrees of freedom from the total system-reservoir density matrix ρtot(t) through a partial trace, ρ(t) = trR{ρtot(t)}.

The resulting dynamics of the open system can be expressed using a quantum master equation of the form dρ(t) dt = L(t)ρ(t), where L(t) is the Liouville von Neumann super-operator, which contains a Hamiltonian part and a dissipative part accounting for the influence of the reservoir. The average energy of the system at an arbitrary time, namely the nonequilibrium internal energy, is given by U(t) = tr{H(t)ρ(t)}.

The first law of nonequilibrium quantum thermodynamics for the open system can be obtained from the rate of change in internal energy, given by dU(t) dt = dW(t) dt + dQ(t) dt, where the first term on the right is the rate of change of work given by dW(t) dt = tr{ H(t)ρ(t)}, and the second term on the right is the rate of heat flow between the system and the reservoir dQ(t) dt = tr{H(t) ρ(t)} = tr{H(t)L(t)ρ(t)}. If the system evolves from time t = t0 to t = τ, the change in internal energy ∆U(τ) = U(τ) −U(t0) will be balanced by the work ∆W(τ) = R τ t0 tr{ H(t)ρ(t)}dt, and heat contribution ∆Q(τ) = R τ t0 tr{H(t)L(t)ρ(t)}dt.

When a quantum system interacts with a thermal reservoir, it may exchange energy, information, or particles. Researchers restrict attention to energy and information exchange in the absence of particle transfer. The decoherence rate or the rate of loss of coherence ΦC(t) is then given by ΦC(t) = −d dtC(t) = −d dtS(ρε(t)) + d dtS(ρ(t)).

Quantum coherence is a fundamental resource for quantum systems, and its quantification has only recently been formalized within a unified resource-theoretic framework. To quantify coherence of a state, they consider the well-known measure of relative entropy of coherence. For any quantum state ρ(t) in nonequilibrium, the relative entropy of coherence is given by C(t) = C(ρ(t)) = min σ(t)∈I S(ρ(t)∥σ(t)), where S(ρ(t)∥σ(t)) = tr [ρ(t) ln ρ(t) −ρ(t) ln σ(t)] measures the distance of ρ(t) from a reference state σ(t).

The reference states σ(t) ∈I, which are incoherent states in the energy eigenbasis. The minimum is evaluated over the set of incoherent states I, which possess no quantum coherence in the energy eigenbasis. It was shown that for the relative entropy of coherence, the closest incoherent state is ρε(t) = P n pn(t)|εn⟩⟨εn|, which is the diagonal state of the density matrix ρ(t) obtained by deleting all off-diagonal elements.

Hence it is not necessary to perform the minimisation to determine the quantum coherence. The relative entropy of coherence is then given by C(t) = S(ρ(t)∥ρε(t)) = S(ρε(t)) −S(ρ(t)), where S(ρ(t)) and S(ρε(t)) are the von Neumann entropy of the state ρ(t) and the diagonal state ρε(t), respectively. Researchers provide the entropy balance relation in complete nonequilibrium, before the system approaches to the steady state or thermal equilibrium.

In this relation, they completely avoid any reference to a thermal equilibrium state. Moreover, the temperature T(t) is not the temperature of the system at thermal equilibrium, rather it is a dynamical temperature evolving with time t. Now, substituting the expressions of ΦQ(t) and ΦC(t) from Eqs. and in Eq., they have Q(t) T(t) = d dtS(ρε(t)).

They emphasize the physical significance of the entropy S(ρε(t)), noting that entropy is fundamentally defined with respect to a probability distribution. For quantum systems, such probabilities arise naturally from measurements of observables. For an observable A = P n an|an⟩⟨an|, a system described by the state ρ(t) yields measurement probabilities Pn(t) = tr (ρ(t)|an⟩⟨an|).

Entropy quantification via diagonalisation of the density matrix and covariance matrix formalism

A central technique in this work involves calculating the probabilities pα0 n (t) of measurement outcomes with respect to an observable H(t). The probability pα0 n (t) is determined by ⟨n|ρ(t)|n⟩, representing the probability of finding the quantum system in a specific energy eigenstate |n⟩ at time t.

Thermodynamic entropy is then defined as S(t) = − Σ n=0 pα0 n (t) ln pα0 n (t), and is experimentally accessible through the probability distribution pα0 n (t). Researchers also evaluated the nonequilibrium von Neumann entropy S(t) of an open quantum system interacting with a thermal reservoir. The system, a single-mode bosonic system initially prepared in a coherent state, is characterised by a Gaussian Wigner function.

For this Gaussian state, the von Neumann entropy is directly evaluated from the covariance matrix V(t), which is fully determined by the first and second moments of the quadrature operators ξ1 = (a + a†) and ξ2 = −i(a −a†). Elements of the covariance matrix are calculated as Vii(t) = ⟨ξ2 i ⟩−⟨ξi⟩2 and Vij(t) = 1/2⟨ξiξj + ξjξi⟩−⟨ξi⟩⟨ξj⟩, utilising averages taken with respect to the nonequilibrium density matrix ρ(t).

The time evolution of quadrature moments was determined using the Heisenberg equation of motion, with detailed calculations provided in supplementary material. Prior to calculating thermodynamic quantities, the entropy balance relation was investigated for this open system model, initially prepared in a coherent state |α0⟩, with the reservoir at thermal equilibrium with kT0 = 20ħω0.

Weak system-reservoir coupling was considered, with η = 0.1ηc, and the time-dependent von Neumann entropy S(t) was computed using Eq. The nonequilibrium thermodynamic entropy S(t) was evaluated through the diagonal density matrix ρε(t), with matrix elements defined by Eq. , representing probabilities pα0 n (t) associated with energy measurement. Finally, entropy production ΦC(t) due to coherence loss and entropy flux ΦQ(t) due to heat exchange were calculated to verify the entropy balance relation Σ(t) = ΦQ(t)+ΦC(t).

Coherence loss and energy entropy define nonequilibrium thermodynamic balances

Researchers have developed a first-principles framework for nonequilibrium thermodynamics by integrating resource theory of coherence with established thermodynamic laws. This work derives a previously unexplored entropy balance relation that distinctly separates entropy flux resulting from heat exchange from entropy production originating from the loss of coherence.

The formulation identifies energy entropy, associated with energy measurements, as the appropriate entropy for nonequilibrium processes, demonstrating that von Neumann entropy does not generally represent entropy away from equilibrium. Within this framework, dynamical temperature, free energy, work, and heat are consistently defined, and both the first and second laws of thermodynamics are shown to remain valid far from equilibrium.

Application of the theory to an exactly solvable open quantum system reveals how equilibrium emerges dynamically in the weak-coupling limit, establishing a unified and operational foundation for nonequilibrium quantum thermodynamics. The study clarifies the fundamental thermodynamic role of quantum coherence in these systems.

The research demonstrates that entropy production in nonequilibrium dynamics arises from two distinct physical origins: heat exchange, causing rearrangement of the system density matrix’s diagonal elements, and information exchange due to the decay of quantum coherence. A key finding is that the conventional identification of thermodynamic entropy with von Neumann entropy is generally unjustified for coherent nonequilibrium states.

This new entropy balance relation provides a quantitative description of the interplay between these two contributions to entropy production. Further analysis using an exactly solvable open quantum system confirms that equilibrium thermodynamics is dynamically recovered as the coupling strength weakens.

The work establishes a foundation for understanding how microscopic quantum dynamics give rise to macroscopic thermodynamic behaviour, particularly in systems exhibiting coherence. This framework offers a novel approach to analysing quantum thermal devices and engines operating far from equilibrium.

Quantum coherence and energy entropy define non-equilibrium thermodynamic relations

Scientists have developed a foundational framework for understanding systems far from thermal equilibrium by integrating quantum coherence with established thermodynamic laws. This new approach derives a previously unknown entropy balance relation that distinguishes between entropy changes caused by heat exchange and those resulting from the dissipation of quantum coherence.

The research demonstrates that conventional von Neumann entropy is not generally representative of entropy when a system is not in equilibrium, instead identifying energy entropy, associated with energy measurements, as the relevant thermodynamic quantity. This formulation consistently defines key thermodynamic concepts such as dynamical temperature, free energy, work, and heat, confirming the validity of both the first and second laws of thermodynamics even in scenarios far from equilibrium.

Application of the theory to a solvable quantum system reveals how equilibrium emerges dynamically as the system’s coupling to its environment weakens. The findings establish a clear connection between quantum coherence, entropy production, and thermodynamic principles, offering a unified basis for nonequilibrium quantum thermodynamics and potentially supporting the development of novel quantum thermal devices.

The authors acknowledge that the framework was applied to a specific, exactly solvable system, representing a limitation in the immediate scope of the results. Future research may broaden the applicability of this approach to more complex, driven, and dissipative quantum systems, further solidifying its role as a foundational tool in the field.