Starting from an algebraic approach of quantum physics it has been shown via the Tomita-Takesaki theorem and the KMS condition that the canonical density matrix contains the dynamics of the system provided we use a rescaling of time. In this paper we show that the path integral form of the partition function reveals a dynamics which is complementary of the one given by the Tomita-Takesaki theorem. To do that we work in the spirit of a Feynman'conjecture. We define the entropy as a measure of the disorder in space time. By using an equilibrium condition we introduce a natural time scale that it is precisely the one appearing in the Tomita-Takesaki theorem. For this time scale depending on the temperature but not on the system properties our definition of entropy is identical to the thermodynamic one. The underlying dynamics associated with the partition function allows us to derive a $\bf{H}$-$theorem$. It is obtained in the thermodynamic limit and provided we are in a regime in which the thermal fluctuations are larger than the quantum ones.
In stochastic thermodynamics standard concepts from macroscopic thermodynamics, such as heat, work, and entropy production, are generalized to small fluctuating systems by defining them on a trajectory-wise level. In Langevin systems with continuous state-space such definitions involve stochastic integrals along system trajectories, whose specific values depend on the discretization rule used to evaluate them (i.e. the "interpretation" of the noise terms in the integral). Via a systematic mathematical investigation of this apparent dilemma, we corroborate the widely used standard interpretatio…
Thermodynamics is independent of a description at a microscopic level consequently statistical thermodynamics must produce results independent of the coordinate system used to describe the particles and their interactions. In the path integral formalism the equilibrium properties are cal- culated by using closed paths and an euclidean coordinate system. We show that the calculations on these paths are coordinates independent. In the change of coordinate systems we consider those preserving the physics on which we focus. Recently it has been shown that the path integral formalism can be built f…
A fundamental problem in modern thermodynamics is how a molecular-scale machine performs useful work, while operating away from thermal equilibrium without excessive dissipation. To this end, we derive a friction tensor that induces a Riemannian manifold on the space of thermodynamic states. Within the linear-response regime, this metric structure controls the dissipation of finite-time transformations, and bestows optimal protocols with many useful properties. We discuss the connection to the existing thermodynamic length formalism, and demonstrate the utility of this metric by solving for op…
Thermodynamic systems involving reversible and non-reversible heat transfer are used to derive integral inequalities expected from the Second of Law of Thermodynamics. Then, the inequalities are proved and generalized to higher dimensions with intended application of being used as a quick way of generating numerous other inequalities such as the Weighted Means Inequality. The proofs also serve as a way of establishing the equivalence of the Planck's statement of the Second Law and the positivity of specific heats.
We consider diffusion-like paths that are explored by a particle moving via a conservative force while being in thermal equilibrium with its surroundings. To probe rare transitions, we use the Onsager-Machlup (OM) functional as a path probability distribution function for double-ended paths that are constrained to start and stop at predesignated points after a fixed time. We explore the continuous-time limit where the OM functional has been commonly regularized by using the Ito-Girsanov change of measure. When used as a path measure, the Ito-Girsanov expression generates an ensemble of double-…
A numerical experiment of ideal stochastic motion of a particle subject to conservative forces and Gaussian noise reveals that the path probability depends exponentially on action. This distribution implies a fundamental principle generalizing the least action principle of the Hamiltonian-Lagrangian mechanics and yields an extended formalism of mechanics for random dynamics. Within this theory, Liouville theorem of conservation of phase density distribution must be modified to allow time evolution of phase density and consequently the Boltzmann H theorem. We argue that the gap between the regu…
Expanding upon previous work, using the path-integral formalism we derive expressions for the one-particle reduced density matrix and the two-point correlation function for a quadratic system of bosons that interact through a general class of memory kernels. The results are applied to study the density, condensate fraction and pair correlation function of trapped bosons harmonically coupled to external distinguishable masses.
Thermodynamic Bethe ansatz equations for XXZ model at $|Δ|\ge 1$ is simplified to an integral equation which has one unknown function. This equation is analytically continued to $|Δ|<1$.
We investigate a thermally isolated quantum many-body system with an external control represented by a step protocol of a parameter. The propagator at each step of the parameter change is described by thermodynamic quantities under some assumptions. For the time evolution of such systems, we formulate a path integral over the trajectories in the thermodynamic state space. In particular, for quasi-static operations, we derive an effective action of the thermodynamic entropy and its canonically conjugate variable. Then, the symmetry for the uniform translation of the conjugate variable emerges i…
For a system of bosons that interact through a class of general memory kernels, a recurrence relation for the partition function is derived within the path-integral formalism. This approach provides a generalization to previously known treatments in the literature of harmonically coupled systems of identical particles. As an example the result is applied to the specific heat of a simplified model of an open quantum system of bosons, harmonically coupled to a reservoir of distinguishable fictitious masses.
The thesis focuses on the prediction of solvation thermodynamics using integral equation theories. Our main goal is to improve the approach using a rational correction. We achieve it by extending recently introduced pressure correction, and rationalizing it in the context of solvation entropy. The improved model (to which we refer as advanced pressure correction) is rather universal. It can accurately predict solvation free energies in water at both ambient and non-ambient temperatures, is capable of addressing ionic solutes and salt solutions, and can be extended to non-aqueous systems. The d…
Work belongs to the most basic notions in thermodynamics but it is not well understood in quantum systems, especially in open quantum systems. By introducing a novel concept of work functional along individual Feynman path, we invent a new approach to study thermodynamics in the quantum regime. Using the work functional, we derive a path-integral expression for the work statistics. By performing the $\hbar$ expansion, we analytically prove the quantum-classical correspondence of the work statistics. In addition, we obtain the quantum correction to the classical fluctuating work. We can also ap…
We study the heat statistics of a quantum Brownian motion described by the Caldeira-Leggett model. By using the path integral approach, we introduce a novel concept of the quantum heat functional along every pair of Feynman paths. This approach has an advantage of improving our understanding about heat in quantum systems. First, we demonstrate the microscopic reversibility of the system by connecting the heat functional to the forward and its time-reversed probabilities. Second, we analytically prove the quantum-classical correspondence of the heat functional and their statistics, which allows…
The paper by Malek Mansour and Garcia [Phys. Rev. E 101, 052135 (2020)] is shown to be based on misconceptions in the stochastic formulation of chemical thermodynamics in reactive systems. Their erroneous claims, asserting that entropy production cannot be correctly evaluated using path probabilities whenever the reactive system involves more than one elementary reaction leading to the same composition changes, are refuted.
We derive a finite set of nonlinear integral equations (NLIE) for the thermodynamics of the one-dimensional sl(4)-symmetric Uimin-Sutherland model. Our NLIE can be evaluated numerically for arbitrary finite temperature and chemical potentials. We recover the NLIE for sl(3) as a limiting case. In comparison to other recently derived NLIE, the evaluation at low temperature poses no problem in our formulation. The model shows a rich ground-state phase diagram. We obtain the critical fields from the T to zero limit of our NLIE. As an example for the application of the NLIE, we give numerical resul…
Thermodynamics is a well developed tool to study systems in equilibrium but no such general framework is available for non-equilibrium processes. Only hope for a quantitative description is to fall back upon the equilibrium language as often done in biology. This gap is bridged by the work theorem. By using this theorem we show that the Barkhausen-type non-equilibrium noise in a process, repeated many times, can be combined to construct a special matrix ${\cal S}$ whose principal eigenvector provides the equilibrium distribution. For an interacting system ${\cal S}$, and hence the equilibrium …
We propose a system of nonlinear integral equations (NLIE), which gives the free energy of the osp(1|2s) integrable spin chain at finite temperatures. In contrast with usual thermodynamic Bethe ansatz equations, our new NLIE contain only a finite number of unknown functions. On deriving NLIE, we use our osp(1|2s) version of the T-system and the quantum transfer matrix method. Based on our NLIE, we also calculate the high temperature expansion of the free energy and the specific heat.
Most recently, the path integral molecular dynamics has been successfully used to consider the thermodynamics of single-component identical bosons and fermions. In this work, the path integral molecular dynamics is developed to simulate the thermodynamics, Green's function and momentum distribution of two-component bosons in three dimensions. As an example of our general method, we consider the thermodynamics of up to sixteen bosons in a three-dimensional harmonic trap. For noninteracting spinor bosons, our simulation shows a bump in the heat capacity. As the repulsive interaction strength inc…
In the general case of a many-body Hamiltonian system, described by an autonomous Hamiltonian $H$, and with $K\geq 0$ independent conserved quantities, we derive the microcanonical thermodynamics. By a simple approach, based on the differential geometry, we derive the microcanonical entropy and the derivatives of the entropy with respect to the conserved quantities. In such a way, we show that all the thermodynamical quantities, as the temperature, the chemical potential or the specific heat, are measured as a microcanonical average of the appropriate microscopic dynamical functions that we ha…
We propose a nonlinear integral equation (NLIE) with only one unknown function, which gives the free energy of the integrable one dimensional Heisenberg model with arbitrary spin. In deriving the NLIE, the quantum Jacobi-Trudi and Giambelli formula (Bazhanov-Reshetikhin formula), which gives the solution of the T-system, plays an important role. In addition, we also calculate the high temperature expansion of the specific heat and the magnetic susceptibility.
I review the generating function for quantum-statistical mechanics, known as the Feynman-Vernon influence functional, the decoherence functional, or the Schwinger-Keldysh path integral. I describe a probability-conserving $i\varepsilon$ prescription from a path-integral implementation of Lindblad evolution. I also explain how to generalize the formalism to accommodate out-of-time-ordered correlators (OTOCs), leading to a Larkin-Ovchinnikov path integral. My goal is to provide step-by-step calculations of path integrals associated to the harmonic oscillator.
We derive traditional thermodynamic Bethe ansatz (TBA) equations for the sl(r+1) Uimin-Sutherland model from the T-system of the quantum transfer matrix. These TBA equations are identical to the ones from the string hypothesis. Next we derive a new family of nonlinear integral equations (NLIE). In particular, a subset of these NLIE forms a system of NLIE which contains only a finite number of unknown functions. For r=1, this subset of NLIE reduces to Takahashi's NLIE for the XXX spin chain. A relation between the traditional TBA equations and our new NLIE is clarified. Based on our new NLIE, w…
We propose a system of nonlinear integral equations (NLIE) which describes the thermodynamics of the U_{q}(\hat{sl(r+1)}) Perk-Schultz model. These NLIE correspond to a trigonometric analogue of our previous result (cond-mat/0212280), and contain only r unknown functions. In particular, they reduce to Takahashi's NLIE for the XXZ spin chain (cond-mat/0010486) if r=1. We also calculate the high temperature expansion of the free energy. In particular for r=1 case, we have succeeded to derive the coefficients of order O((\frac{J}{T})^{99}).
Path Integral Methods in Atomistic Modelling: An Introduction
Institut des Matériaux, École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院材料研究所）；Department of Chemistry, University of Oxford（牛津大学化学系）；Department of Chemistry, Stanford University（斯坦福大学化学系）；MPI for the Structure and Dynamics of Matter（马普物质结构与动力学研究所）；Yusuf Hamied Chemistry Department, University of Cambridge（剑桥大学尤瑟夫·哈米德化学系）
Stochastic mechanics---the study of classical stochastic systems governed by things like master equations and Fokker-Planck equations---exhibits striking mathematical parallels to quantum mechanics. In this article, we make those parallels more transparent by presenting a quantum mechanics-like formalism for deriving a path integral description of systems described by stochastic differential equations. Our formalism expediently recovers the usual path integrals (the Martin-Siggia-Rose-Janssen-De Dominicis and Onsager-Machlup forms) and is flexible enough to account for different variable domai…
Experimental realizations of a variety of atomic binary Bose-Fermi mixtures have brought opportunities for studying composite quantum systems with different spin-statistics. The binary atomic mixtures can exhibit a structural transition from a mixture into phase separation as the boson-fermion interaction increases. By using a path-integral formalism to evaluate the grand partition function and thermodynamic grand potential, we obtain the effective potential of binary Bose-Fermi mixtures. Thermodynamic quantities in a broad range of temperatures and interactions are also derived. The structura…
We formulate Bayesian updates in Markov processes by means of path integral techniques and derive the imaginary-time Schrödinger equation with likelihood to direct the inference incorporated as a potential for the posterior probability distribution
We derive the relativistic thermodynamic scale equation using imaginary-time path integrals, with complex scalar field theory taken as a concrete example. We use Fujikawa's method to derive the scaling anomaly for this system using a matrix regulator. We make a general scaling argument to show how for anomalous systems, the $β$ function of the vacuum theory can be derived from measurement of macroscopic thermodynamic parameters.
In this paper I develop a new computational method for pricing path dependent options. Using the path integral representation of the option price, I show that in general it is possible to perform analytically a partial averaging over the underlying risk-neutral diffusion process. This result greatly eases the computational burden placed on the subsequent numerical evaluation. For short-medium term options it leads to a general approximation formula that only requires the evaluation of a one dimensional integral. I illustrate the application of the method to Asian options and occupation time de…
A recently developed analytical method for systematic improvement of the convergence of path integrals is used to derive a generalization of Euler's summation formula for path integrals. The first $p$ terms in this formula improve convergence of path integrals to the continuum limit from 1/N to $1/N^p$, where $N$ is the coarseness of the discretization. Monte Carlo simulations performed on several different models show that the analytically derived speedup holds.
