Hamiltonian Of Mean Force Research Articles

Thermodynamics and stochastic thermodynamics of strongly coupled systems.

We further develop the strong-coupling theory of thermodynamics and stochastic thermodynamics for continuous systems, constructed in the previous work [Phys. Rev. Res. 4, 013015 (2022)2643-156410.1103/PhysRevResearch.4.013015]. A small system strongly interacting with a its environment, the dynamics of the system is assumed to be much slower than that of the bath. The system Hamiltonian is defined to be the Hamiltonian of mean force, whereas the system entropy is defined as the Gibbs-Shannon entropy. Equilibrium ensemble theories and thermodynamic theories are established for the system. Variations of three types of parameters are considered: (i) the system parameter λ which couples to the system and to the interaction, (ii) the bath parameter λ^{'} which couples to the bath only, and (iii) the temperature T=1/β. The work done to the system consists of three parts, proportional to dλ,dλ^{'}, and dβ respectively. The part proportional to dβ can be understood as the work done by the bath. As long as λ^{'} and β are not fixed, the work is not the change of total energy of the joint system. The differences between our strong-coupling equilibrium thermodynamics and the classical thermodynamics are discussed. The thermodynamic theory is promoted to the nonequilibrium level. Both the first and second laws of thermodynamics, as well as fluctuation theorems, are established for nonequilibrium processes. For processes with varying temperatures, fluctuation theorems cannot be expressed in terms of integrated work alone. Regardless of various subtleties, however, the stochastic thermodynamic theory is formulated in terms of system variables only, and dS-βd[over ¯]Q is the change of total entropy. Thermodynamic quantities of the system are related to those of the joint system, and the equivalence of theories at two levels of coarse-graining is explicitly demonstrated. Finally we show that there are infinite numbers of equivalent strong-coupling theories, each determined by its definition of system Hamiltonian. Our theory is distinguished by its maximal similarity with the weak-coupling theory.

Hamiltonian of mean force in the weak-coupling and high-temperature approximations and refined quantum master equations

The Hamiltonian of mean force is a widely used concept to describe the modification of the usual canonical Gibbs state for a quantum system whose coupling strength with the thermal bath is non-negligible. Here, we perturbatively derive general approximate expressions for the Hamiltonians of mean force in the weak-coupling approximation and in the high-temperature one. We numerically analyze the accuracy of the corresponding expressions and show that the precision of the Bloch–Redfield equantum master equation can be improved if we replace the original system Hamiltonian by the Hamiltonian of mean force.

Numerical evaluation and robustness of the quantum mean-force Gibbs state

We introduce a numerical method to determine the Hamiltonian of mean force (HMF) Gibbs state for a quantum system strongly coupled to a reservoir. The method adapts the time evolving matrix product operator (TEMPO) algorithm to imaginary-time propagation. By comparing the real-time and imaginary-time propagation for a generalized spin-boson model, we confirm that the HMF Gibbs state correctly predicts the steady state. We show that the numerical dynamics match the polaron master equation at strong coupling. We illustrate the potential of the imaginary-time TEMPO approach by exploring reservoir-induced entanglement between qubits.

Strong coupling thermodynamics and stochastic thermodynamics from the unifying perspective of time-scale separation

Assuming time-scale separation, a simple and unified theory of thermodynamics and stochastic thermodynamics is constructed for small classical systems strongly interacting with their environments in a controllable fashion. The total Hamiltonian is decomposed into a bath part and a system part, the latter being the Hamiltonian of mean force. Both the conditional equilibrium of the bath and the reduced equilibrium of the system are described by canonical ensemble theories with respect to their own Hamiltonians. The bath free energy is independent of the system variables and the control parameter. Furthermore, the weak coupling theory of stochastic thermodynamics becomes applicable almost verbatim, even if the interaction and correlation between the system and its environment are strong and varied externally. We further discuss a simple scenario where the present theory fits better with the common intuition about system entropy and heat.

Statistical Mechanics at Strong Coupling: A Bridge between Landsberg's Energy Levels and Hill's Nanothermodynamics.

We review and show the connection between three different theories proposed for the thermodynamic treatment of systems not obeying the additivity ansatz of classical thermodynamics. In the 1950s, Landsberg proposed that when a system comes into contact with a heat bath, its energy levels are redistributed. Based on this idea, he produced an extended thermostatistical framework that accounts for unknown interactions with the environment. A decade later, Hill devised his celebrated nanothermodynamics, where he introduced the concept of subdivision potential, a new thermodynamic variable that accounts for the vanishing additivity of increasingly smaller systems. More recently, a thermostatistical framework at strong coupling has been formulated to account for the presence of the environment through a Hamiltonian of mean force. We show that this modified Hamiltonian yields a temperature-dependent energy landscape as earlier suggested by Landsberg, and it provides a thermostatistical foundation for the subdivision potential, which is the cornerstone of Hill’s nanothermodynamics.

Hamiltonian of Mean Force and Dissipative Scalar Field Theory

Quantum dynamics of a dissipative scalar field is investigated. Using the Hamiltonian of mean force, internal energy, free energy and entropy of a dissipative scalar field are obtained. It is shown that a dissipative massive scalar field can be considered as a free massive scalar field described by an effective mass and dispersion relation. Internal energy of the scalar field, as the subsystem, is found in the limit of low temperature and weak and strong couplings to an Ohimc heat bath. Correlation functions for thermal and coherent states are derived.

Entropy production and time asymmetry in the presence of strong interactions.

It is known that the equilibrium properties of open classical systems that are strongly coupled to a heat bath are described by a set of thermodynamic potentials related to the system's Hamiltonian of mean force. By adapting this framework to a more general class of nonequilibrium states, we show that the equilibrium properties of the bath can be well defined, even when the system is arbitrarily far from equilibrium and correlated with the bath. These states, which retain a notion of temperature, take the form of conditional equilibrium distributions. For out-of-equilibrium processes we show that the average entropy production quantifies the extent to which the system and bath state is driven away from the conditional equilibrium distribution. In addition, we show that the stochastic entropy production satisfies a generalized Crooks relation and can be used to quantify time asymmetry of correlated nonequilibrium processes. These results naturally extend the familiar properties of entropy production in weakly coupled systems to the strong coupling regime. Experimental measurements of the entropy production at strong coupling could be pursued using optomechanics or trapped-ion systems, which allow strong coupling to be engineered.

Stochastic thermodynamics in the strong coupling regime: An unambiguous approach based on coarse graining.

We consider a classical and possibly driven composite system X⊗Y weakly coupled to a Markovian thermal reservoir R so that an unambiguous stochastic thermodynamics ensues for X⊗Y. This setup can be equivalently seen as a system X strongly coupled to a non-Markovian reservoir Y⊗R. We demonstrate that only in the limit where the dynamics of Y is much faster than X, our unambiguous expressions for thermodynamic quantities, such as heat, entropy, or internal energy, are equivalent to the strong coupling expressions recently obtained in the literature using the Hamiltonian of mean force. By doing so, we also significantly extend these results by formulating them at the level of instantaneous rates and by allowing for time-dependent couplings between X and its environment. Away from the limit where Y evolves much faster than X, previous approaches fail to reproduce the correct results from the original unambiguous formulation, as we illustrate numerically for an underdamped Brownian particle coupled strongly to a non-Markovian reservoir.

Hamiltonian of mean force and a damped harmonic oscillator in an anisotropic medium

The quantum dynamics of a damped harmonic oscillator is investigated in the presence of an anisotropic heat bath. The medium is modeled by a continuum of three dimensional harmonic oscillators and anisotropic coupling is treated by introducing tensor coupling functions. Starting from a classical Lagrangian, the total system is quantized in the framework of the canonical quantization. Following the Fano technique, the Hamiltonian of the system is diagonalized in terms of creation and annihilation operators that are linear combinations of the basic dynamical variables. Using the diagonalized Hamiltonian, the mean force internal energy, free energy and entropy of the damped oscillator are calculated.