Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0902.0931

Relation Time-Thermodynamics. a Path Integral Approach

J. P. Badiali

Published Feb 5, 2009• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.0902.0931

Related papers

Towards a new approach of quantum dissipation in simple chemical systems

J. P. Badiali

We suggest a new approach for describing quantum dissipation in a small systems for which the system-plus-reservoir approach is not relevant. We first analyze the fact that equilibrium thermodynamics may reveal the existence of an underlying dynamics. This is true in the algebraic approach of quantum mechanics via the Tomita-Takesaki theorem. A similar result is obtained if we start from the path integral expression of the partition function, an equation of motion is introduced. In both cases a parameter has to be identified with the physical time. Several arguments are presented showing that it is so. By investigating the dynamics for short times we introduce an equilibrium condition from which a natural unit of time appears. The equation of motion is extended to non equilibrium situations for which a H-theorem can be derived. The equation of motion can be transformed into a quantum Smoluchovski equation, its solution is a probability having a clear physical meaning. The relaxation of this probability towards its equilibrium value requires a time during which a thermodynamic description is possible or not. A standard bistable model is investigated and the chemical rate $k(t)$ is calculated as a function of time, it appears to be a non monotonic function of time. In some conditions the stationary value of $k(t)$ can be identified with the Kramers result. Finally we compare our approach which is naturally irreversible with an analysis based on the Schrödinger equation which is reversible. From all these results it appears that our equation of motion or its Smoluchovski version appear as a realistic starting point for describing quantum dissipation in small systems.

arXiv:0910.1948•Oct 12, 2009•0 citations

FRACTAL BEHAVIOUR OF QUANTUM PATHS IN STATISTICAL PHYSICS

J. P. Badiali

The path integral formalism is used to describe the statistical properties of an ideal gas of spinless particles.It is shown that the quantum paths exhibit the same properties in non-relativistic and relativistic domains provided the creation of new particles is avoided.Some quantities associated with the paths are introduced, they have a simple meaning if the quantity β , where β is the reverse of the temperature, is considered as an ordinary time.The relation between the velocity on the path and the momentum is not the usual one, an extra term appears showing that the thermostat can not fix the average value of this velocity although all the thermodynamic quantities have their traditional values.The paths describe fluctuating trajectories on which the particles do not follow the equation of motion.For time intervals much shorter than β we recover the properties of the Brownian motion.The trajectories are located in space in a volume restricted by the Compton wavelength for the short distances and the thermal de Broglie wavelength for the largest ones.It is shown that the time-energy uncertainty is verified on the quantum paths.This suggests that the density matrix obtained by quantification of the classical canonical distribution function via the path integral formalism should not be totally identical to that obtained via the usual route.Strong arguments are given showing that β can be considered as an ordinary time and not as a formal quantity having the same dimension as time.This paper shows that for a time scale of 10 femtoseconds a totally new physics can be expected at room temperature.In addition it is suggested that the ratio /k B may play a decisive role in the foundation of a covariant statistical physics.

Condensed Matter Physics•Jan 1, 2000•5 citations

Quantum Statistical Mechanics Functional Integral Approach

Lukong Cornelius Fai

We establish a rather deep relation between statistical mechanics and quantum mechanics at an imaginary time. For the quantum field theory, this implies a relation between finite temperature quantum field theory in Minkowski space and quantum field theory in Euclidean space with one compact Euclidean time direction. The quantum mechanical language being a proper formulation of statistical physics as a molecular theory is basically the study of quantum equilibrium as well as non-equilibrium systems. A quantum system found in a thermodynamic equilibrium state with an external classical system (thermostat or thermal reservoir) may be characterized by fixed values of thermodynamic parameters of the system, say, for example, temperature T , volume V and number of particles N . In this way the total energy of the thermostat is assumed to be much greater than the energy of the system. The behavior of the quantum system may be assumed to be independent of the nature of the thermostat. The most convenient object of the quantum theory to describe such non-isolated systems is the density matrix since the partition function does not determine any local thermodynamic quantities. It is for this reason that we introduced quantum mechanical and quantum statistical density matrices which are necessary to do the investigation via path integration. This enables us to make a transition to the quasi-classical approximation of quantum statistical mechanics, which, in turn, enables us to evaluate the first quantum correction to the classical Boltzmann distribution function.

CRC Press eBooks•Feb 19, 2021•0 citations

Fractal behaviour of quantum paths in statistical physics

J. P. Badiali

The path integral formalism is used to describe the statistical properties of \nan ideal gas of spinless particles. It is shown that the quantum paths exhibit \nthe same properties in non-relativistic and relativistic domains provided the \ncreation of new particles is avoided. Some quantities associated with the \npaths are introduced, they have a simple meaning if the quantity βh , where \nβ is the reverse of the temperature, is considered as an ordinary time. The \nrelation between the velocity on the path and the momentum is not the \nusual one, an extra term appears showing that the thermostat can not fix \nthe average value of this velocity although all the thermodynamic quantities \nhave their traditional values. The paths describe fluctuating trajectories on \nwhich the particles do not follow the equation of motion. For time intervals \nmuch shorter than βh we recover the properties of the Brownian motion. \nThe trajectories are located in space in a volume restricted by the Compton \nwavelength for the short distances and the thermal de Broglie wavelength \nfor the largest ones. It is shown that the time-energy uncertainty is verified \non the quantum paths. This suggests that the density matrix obtained by \nquantification of the classical canonical distribution function via the path integral formalism should not be totally identical to that obtained via the usual \nroute. Strong arguments are given showing that βh can be considered as \nan ordinary time and not as a formal quantity having the same dimension \nas time. This paper shows that for a time scale of 10 femtoseconds a totally new physics can be expected at room temperature. In addition it is \nsuggested that the ratio h/kB may play a decisive role in the foundation of \na covariant statistical physics.

•Jan 1, 2000•0 citations

Fractal behavior in quantum statistical physics

J. P. Badiali

The properties of an ideal gas of spinless particles are investigated by using the path integral formalism. It is shown that the quantum paths exhibit a fractal character which remains unchanged in the relativistic domain provided the creation of new particles is avoided, and the Brownian motion remains the stochastic process associated with the quantum paths. These results are obtained by using a special representation of the Klein-Gordon wave equation. On the quantum paths the relation between velocity and momentum is not the usual one. The mean square value of the velocity depends on the time needed to define the velocity and its value shows the interplay between pure quantum effects and thermodynamics. The fractal character is also investigated starting from wave equations by analyzing the evolution of a Gaussian wave packet via the Hausdorff dimension. Both approaches give the same fractal character in the same limit. It is shown that the time that appears in the path integral behaves like an ordinary time, and the key quantity is the time interval needed for the thermostat to give to the particles a thermal action equal to the quantum of action. Thus, the partition function calculated via the path integral formalism also describes the dynamics of the system for short time intervals. For low temperatures, it is shown that a time-energy uncertainty relation is verified at the end of the calculations. The energy involved in this relation has not a thermodynamic meaning but results from the fact that the particles do not follow the equations of motion along the paths. The results suggest that the density matrix obtained by quantification of the classical canonical distribution function via the path integral formalism should not be totally identical to that obtained via the usual route.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Sep 1, 1999•11 citations

Thermalisation of a quantum system from first principles

Grégoire Ithier, Florent Benaych-Georges

Why is thermalisation a universal phenomenon? How does a quantum system reach thermodynamical equilibrium? These questions are not new, dating even from the very birth of quantum theory and have been the subject of a renewed interest over the two last decades (see for instance the review in Eisert et al, Nature Physics 2015). In this work, we propose a universal model demonstrating that thermalisation of a small quantum system is an emergent property of the unitary evolution under a Schrödinger equation of a larger composite system, whose initial state can be arbitrary. We show that the origin of universality lies in the phenomenon of 'measure concentration', which provides self-averaging properties for the reduced density matrix characterizing the state of the small subsystem. Using our framework, we focus on the asymptotic state at long times and consider its stationary properties. In typical macroscopic conditions, we recover the canonical state and the Boltzmann distribution well known from statistical thermodynamics. This findings lead us to propose an alternative and more general definition of the canonical partition function which also allow us to describe non thermal stationary states.

arXiv:1510.04352•Oct 14, 2015•3 citations
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 H\bf{H}H-theoremtheoremtheorem. 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.
We suggest a new approach for describing quantum dissipation in a small systems for which the system-plus-reservoir approach is not relevant. We first analyze the fact that equilibrium thermodynamics may reveal the existence of an underlying dynamics. This is true in the algebraic approach of quantum mechanics via the Tomita-Takesaki theorem. A similar result is obtained if we start from the path integral expression of the partition function, an equation of motion is introduced. In both cases a parameter has to be identified with the physical time. Several arguments are presented showing that it is so. By investigating the dynamics for short times we introduce an equilibrium condition from which a natural unit of time appears. The equation of motion is extended to non equilibrium situations for which a H-theorem can be derived. The equation of motion can be transformed into a quantum Smoluchovski equation, its solution is a probability having a clear physical meaning. The relaxation of this probability towards its equilibrium value requires a time during which a thermodynamic description is possible or not. A standard bistable model is investigated and the chemical rate k(t)k(t)k(t) is calculated as a function of time, it appears to be a non monotonic function of time. In some conditions the stationary value of k(t)k(t)k(t) can be identified with the Kramers result. Finally we compare our approach which is naturally irreversible with an analysis based on the Schrödinger equation which is reversible. From all these results it appears that our equation of motion or its Smoluchovski version appear as a realistic starting point for describing quantum dissipation in small systems.
The path integral formalism is used to describe the statistical properties of an ideal gas of spinless particles.It is shown that the quantum paths exhibit the same properties in non-relativistic and relativistic domains provided the creation of new particles is avoided.Some quantities associated with the paths are introduced, they have a simple meaning if the quantity β , where β is the reverse of the temperature, is considered as an ordinary time.The relation between the velocity on the path and the momentum is not the usual one, an extra term appears showing that the thermostat can not fix the average value of this velocity although all the thermodynamic quantities have their traditional values.The paths describe fluctuating trajectories on which the particles do not follow the equation of motion.For time intervals much shorter than β we recover the properties of the Brownian motion.The trajectories are located in space in a volume restricted by the Compton wavelength for the short distances and the thermal de Broglie wavelength for the largest ones.It is shown that the time-energy uncertainty is verified on the quantum paths.This suggests that the density matrix obtained by quantification of the classical canonical distribution function via the path integral formalism should not be totally identical to that obtained via the usual route.Strong arguments are given showing that β can be considered as an ordinary time and not as a formal quantity having the same dimension as time.This paper shows that for a time scale of 10 femtoseconds a totally new physics can be expected at room temperature.In addition it is suggested that the ratio /k B may play a decisive role in the foundation of a covariant statistical physics.
We establish a rather deep relation between statistical mechanics and quantum mechanics at an imaginary time. For the quantum field theory, this implies a relation between finite temperature quantum field theory in Minkowski space and quantum field theory in Euclidean space with one compact Euclidean time direction. The quantum mechanical language being a proper formulation of statistical physics as a molecular theory is basically the study of quantum equilibrium as well as non-equilibrium systems. A quantum system found in a thermodynamic equilibrium state with an external classical system (thermostat or thermal reservoir) may be characterized by fixed values of thermodynamic parameters of the system, say, for example, temperature T , volume V and number of particles N . In this way the total energy of the thermostat is assumed to be much greater than the energy of the system. The behavior of the quantum system may be assumed to be independent of the nature of the thermostat. The most convenient object of the quantum theory to describe such non-isolated systems is the density matrix since the partition function does not determine any local thermodynamic quantities. It is for this reason that we introduced quantum mechanical and quantum statistical density matrices which are necessary to do the investigation via path integration. This enables us to make a transition to the quasi-classical approximation of quantum statistical mechanics, which, in turn, enables us to evaluate the first quantum correction to the classical Boltzmann distribution function.
The path integral formalism is used to describe the statistical properties of \nan ideal gas of spinless particles. It is shown that the quantum paths exhibit \nthe same properties in non-relativistic and relativistic domains provided the \ncreation of new particles is avoided. Some quantities associated with the \npaths are introduced, they have a simple meaning if the quantity βh , where \nβ is the reverse of the temperature, is considered as an ordinary time. The \nrelation between the velocity on the path and the momentum is not the \nusual one, an extra term appears showing that the thermostat can not fix \nthe average value of this velocity although all the thermodynamic quantities \nhave their traditional values. The paths describe fluctuating trajectories on \nwhich the particles do not follow the equation of motion. For time intervals \nmuch shorter than βh we recover the properties of the Brownian motion. \nThe trajectories are located in space in a volume restricted by the Compton \nwavelength for the short distances and the thermal de Broglie wavelength \nfor the largest ones. It is shown that the time-energy uncertainty is verified \non the quantum paths. This suggests that the density matrix obtained by \nquantification of the classical canonical distribution function via the path integral formalism should not be totally identical to that obtained via the usual \nroute. Strong arguments are given showing that βh can be considered as \nan ordinary time and not as a formal quantity having the same dimension \nas time. This paper shows that for a time scale of 10 femtoseconds a totally new physics can be expected at room temperature. In addition it is \nsuggested that the ratio h/kB may play a decisive role in the foundation of \na covariant statistical physics.
The properties of an ideal gas of spinless particles are investigated by using the path integral formalism. It is shown that the quantum paths exhibit a fractal character which remains unchanged in the relativistic domain provided the creation of new particles is avoided, and the Brownian motion remains the stochastic process associated with the quantum paths. These results are obtained by using a special representation of the Klein-Gordon wave equation. On the quantum paths the relation between velocity and momentum is not the usual one. The mean square value of the velocity depends on the time needed to define the velocity and its value shows the interplay between pure quantum effects and thermodynamics. The fractal character is also investigated starting from wave equations by analyzing the evolution of a Gaussian wave packet via the Hausdorff dimension. Both approaches give the same fractal character in the same limit. It is shown that the time that appears in the path integral behaves like an ordinary time, and the key quantity is the time interval needed for the thermostat to give to the particles a thermal action equal to the quantum of action. Thus, the partition function calculated via the path integral formalism also describes the dynamics of the system for short time intervals. For low temperatures, it is shown that a time-energy uncertainty relation is verified at the end of the calculations. The energy involved in this relation has not a thermodynamic meaning but results from the fact that the particles do not follow the equations of motion along the paths. The results suggest that the density matrix obtained by quantification of the classical canonical distribution function via the path integral formalism should not be totally identical to that obtained via the usual route.
Why is thermalisation a universal phenomenon? How does a quantum system reach thermodynamical equilibrium? These questions are not new, dating even from the very birth of quantum theory and have been the subject of a renewed interest over the two last decades (see for instance the review in Eisert et al, Nature Physics 2015). In this work, we propose a universal model demonstrating that thermalisation of a small quantum system is an emergent property of the unitary evolution under a Schrödinger equation of a larger composite system, whose initial state can be arbitrary. We show that the origin of universality lies in the phenomenon of 'measure concentration', which provides self-averaging properties for the reduced density matrix characterizing the state of the small subsystem. Using our framework, we focus on the asymptotic state at long times and consider its stationary properties. In typical macroscopic conditions, we recover the canonical state and the Boltzmann distribution well known from statistical thermodynamics. This findings lead us to propose an alternative and more general definition of the canonical partition function which also allow us to describe non thermal stationary states.