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Gregory A Voth - One of the best experts on this subject based on the ideXlab platform.

  • a reductionist perspective on Quantum Statistical Mechanics coarse graining of path integrals
    Journal of Chemical Physics, 2015
    Co-Authors: Anton V Sinitskiy, Gregory A Voth
    Abstract:

    Computational modeling of the condensed phase based on classical Statistical Mechanics has been rapidly developing over the last few decades and has yielded important information on various systems containing up to millions of atoms. However, if a system of interest contains important Quantum effects, well-developed classical techniques cannot be used. One way of treating finite temperature Quantum systems at equilibrium has been based on Feynman’s imaginary time path integral approach and the ensuing Quantum-classical isomorphism. This isomorphism is exact only in the limit of infinitely many classical quasiparticles representing each physical Quantum particle. In this work, we present a reductionist perspective on this problem based on the emerging methodology of coarse-graining. This perspective allows for the representations of one Quantum particle with only two classical-like quasiparticles and their conjugate momenta. One of these coupled quasiparticles is the centroid particle of the Quantum path integral quasiparticle distribution. Only this quasiparticle feels the potential energy function. The other quasiparticle directly provides the observable averages of Quantum mechanical operators. The theory offers a simplified perspective on Quantum Statistical Mechanics, revealing its most reductionist connection to classical Statistical physics. By doing so, it can facilitate a simpler representation of certain Quantum effects in complex molecular environments.

  • the formulation of Quantum Statistical Mechanics based on the feynman path centroid density iii phase space formalism and analysis of centroid molecular dynamics
    Journal of Chemical Physics, 1994
    Co-Authors: Jianshu Cao, Gregory A Voth
    Abstract:

    The formulation of Quantum Statistical Mechanics based on the path centroid variable in Feynman path integration is generalized to a phase space perspective, thereby including the momentum as an independent dynamical variable. By virtue of this approach, operator averages and imaginary time correlation functions can be expressed in terms of an averaging over the multidimensional phase space centroid density. The imaginary time centroid‐constrained correlation function matrix for the phase space variables is then found to define the effective thermal width of the phase space centroid variable. These developments also make it possible to rigorously analyze the centroid molecular dynamics method for computing Quantum dynamical time correlation functions. As a result, the centroid time correlation function as calculated from centroid molecular dynamics is shown to be a well‐defined approximation to the exact Kubo transformed position correlation function. This analysis thereby clarifies the underlying role of the equilibrium path centroid variable in the Quantum dynamical position correlation function and provides a sound theoretical basis for the centroid molecular dynamics method.

  • the formulation of Quantum Statistical Mechanics based on the feynman path centroid density ii dynamical properties
    Journal of Chemical Physics, 1994
    Co-Authors: Jianshu Cao, Gregory A Voth
    Abstract:

    The formulation of Quantum dynamical time correlation functions is examined within the context of the path centroid variable in Feynman path integration. This study builds on the centroid‐based approach to equilibrium properties developed in the companion paper. The introduction of the centroid perspective into the calculation of real time position correlation functions is outlined and an intriguing quasiclassical role for the centroid variable in real time position correlation functions is identified. This quasiclassical perspective is developed in terms of general interaction potentials, and the computational effort in implementing the method should scale with the size of the system in the same fashion as a classical molecular dynamics calculation. The centroid‐based theory is also implemented in several different approaches to calculate general time correlation functions. The theoretical results are illustrated and tested by representative numerical applications.

Jianshu Cao - One of the best experts on this subject based on the ideXlab platform.

  • the formulation of Quantum Statistical Mechanics based on the feynman path centroid density iii phase space formalism and analysis of centroid molecular dynamics
    Journal of Chemical Physics, 1994
    Co-Authors: Jianshu Cao, Gregory A Voth
    Abstract:

    The formulation of Quantum Statistical Mechanics based on the path centroid variable in Feynman path integration is generalized to a phase space perspective, thereby including the momentum as an independent dynamical variable. By virtue of this approach, operator averages and imaginary time correlation functions can be expressed in terms of an averaging over the multidimensional phase space centroid density. The imaginary time centroid‐constrained correlation function matrix for the phase space variables is then found to define the effective thermal width of the phase space centroid variable. These developments also make it possible to rigorously analyze the centroid molecular dynamics method for computing Quantum dynamical time correlation functions. As a result, the centroid time correlation function as calculated from centroid molecular dynamics is shown to be a well‐defined approximation to the exact Kubo transformed position correlation function. This analysis thereby clarifies the underlying role of the equilibrium path centroid variable in the Quantum dynamical position correlation function and provides a sound theoretical basis for the centroid molecular dynamics method.

  • the formulation of Quantum Statistical Mechanics based on the feynman path centroid density ii dynamical properties
    Journal of Chemical Physics, 1994
    Co-Authors: Jianshu Cao, Gregory A Voth
    Abstract:

    The formulation of Quantum dynamical time correlation functions is examined within the context of the path centroid variable in Feynman path integration. This study builds on the centroid‐based approach to equilibrium properties developed in the companion paper. The introduction of the centroid perspective into the calculation of real time position correlation functions is outlined and an intriguing quasiclassical role for the centroid variable in real time position correlation functions is identified. This quasiclassical perspective is developed in terms of general interaction potentials, and the computational effort in implementing the method should scale with the size of the system in the same fashion as a classical molecular dynamics calculation. The centroid‐based theory is also implemented in several different approaches to calculate general time correlation functions. The theoretical results are illustrated and tested by representative numerical applications.

Matilde Marcolli - One of the best experts on this subject based on the ideXlab platform.

  • Quantum Statistical Mechanics in arithmetic topology
    Journal of Geometry and Physics, 2017
    Co-Authors: Matilde Marcolli
    Abstract:

    This paper provides a construction of a Quantum Statistical mechanical system associated to knots in the 33-sphere and cyclic branched coverings of the 33-sphere, which is an analog, in the sense of arithmetic topology, of the Bost–Connes system, with knots replacing primes, and cyclic branched coverings of the 33-sphere replacing abelian extensions of the field of rational numbers. The operator algebraic properties of this system differ significantly from the Bost–Connes case, due to the properties of the action of the semigroup of knots on a direct limit of knot groups. The resulting algebra of observables is a noncommutative Bernoulli product. We describe the main properties of the associated Quantum Statistical mechanical system and of the relevant partition functions, which are obtained from simple knot invariants like genus and crossing number.

  • bost connes systems categorification Quantum Statistical Mechanics and weil numbers
    Journal of Noncommutative Geometry, 2017
    Co-Authors: Matilde Marcolli, Goncalo Tabuada
    Abstract:

    In this article we develop a broad generalization of the classical Bost–Connes system, where roots of unity are replaced by an algebraic datum consisting of an abelian group and a semi-group of endomorphisms. Examples include roots of unity, Weil restriction, algebraic numbers,Weil numbers, CM fields, germs, completion ofWeil numbers, etc. Making use of the Tannakian formalism, we categorify these algebraic data. For example, the categorification of roots of unity is given by a limit of orbit categories of Tate motives while the categorification of Weil numbers is given by Grothendieck’s category of numerical motives over a finite field. To some of these algebraic data (e.g. roots of unity, algebraic numbers, Weil numbers, etc), we associate also a Quantum Statistical mechanical system with several remarkable properties, which generalize those of the classical Bost–Connes system. The associated partition function, low temperature Gibbs states, and Galois action on zero-temperature states are then studied in detail. For example, we show that in the particular case of the Weil numbers the partition function and the low temperature Gibbs states can be described as series of polylogarithms.

  • bost connes systems categorification Quantum Statistical Mechanics and weil numbers
    arXiv: Mathematical Physics, 2014
    Co-Authors: Matilde Marcolli, Goncalo Tabuada
    Abstract:

    In this article we develop a broad generalization of the classical Bost-Connes system, where roots of unit are replaced by an algebraic datum consisting of an abelian group and a semi-group of endomorphisms. Examples include roots of unit, Weil restriction, algebraic numbers, Weil numbers, CM fields, germs, completion of Weil numbers, etc. Making use of the Tannakian formalism, we categorify these algebraic data. For example, the categorification of roots of unit is given by a limit of orbit categories of Tate motives while the categorification of Weil numbers is given by Grothendieck's category of numerical motives over a finite field. To some of these algebraic data (roots of unity, algebraic numbers, Weil numbers, etc), we associate also a Quantum Statistical mechanical system with several remarkable properties, which generalize those of the classical Bost-Connes system. The associated partition function, low temperature Gibbs states, and Galois action on zero-temperature states are then studied in detail. For example, we show that in the particular case of the Weil numbers the partition function and the low temperature Gibbs states can be described as series of polylogarithms.

  • Quantum Statistical Mechanics l series and anabelian geometry i partition functions
    Trends in Contemporary Mathematics, 2014
    Co-Authors: Gunther Cornelissen, Matilde Marcolli
    Abstract:

    The zeta function of a number field can be interpreted as the partition function of an associated Quantum Statistical mechanical (QSM) system, built from abelian class field theory.

  • graph reconstruction and Quantum Statistical Mechanics
    Journal of Geometry and Physics, 2013
    Co-Authors: Gunther Cornelissen, Matilde Marcolli
    Abstract:

    We study how far it is possible to reconstruct a graph from various Banach algebras associated to its universal covering, and extensions thereof to Quantum Statistical mechanical systems. It turns out that most the boundary operator algebras reconstruct only topological information, but the Statistical mechanical point of view allows for complete reconstruction of multigraphs with minimal degree three.

Claudealain Pillet - One of the best experts on this subject based on the ideXlab platform.

  • a note on the landauer principle in Quantum Statistical Mechanics
    Journal of Mathematical Physics, 2014
    Co-Authors: Vojkan Jaksic, Claudealain Pillet
    Abstract:

    The Landauer principle asserts that the energy cost of erasure of one bit of information by the action of a thermal reservoir in equilibrium at temperature T is never less than kT log 2. We discuss Landauer's principle for Quantum Statistical models describing a finite level Quantum system S coupled to an infinitely extended thermal reservoir R. Using Araki's perturbation theory of KMS states and the Avron-Elgart adiabatic theorem we prove, under a natural ergodicity assumption on the joint system S+R, that Landauer's bound saturates for adiabatically switched interactions. The recent work of Reeb and Wolf on the subject is discussed and compared.

  • Quantum hypothesis testing and non equilibrium Statistical Mechanics
    Reviews in Mathematical Physics, 2012
    Co-Authors: Vojkan Jaksic, Yoshiko Ogata, Claudealain Pillet, Robert Seiringer
    Abstract:

    We extend the mathematical theory of Quantum hypothesis testing to the general W*-algebraic setting and explore its relation with recent developments in non-equilibrium Quantum Statistical Mechanics. In particular, we relate the large deviation principle for the full counting statistics of entropy flow to Quantum hypothesis testing of the arrow of time.

  • entropic fluctuations in Quantum Statistical Mechanics an introduction
    arXiv: Mathematical Physics, 2011
    Co-Authors: Vojkan Jaksic, Yoshiko Ogata, Yan Pautrat, Claudealain Pillet
    Abstract:

    These lecture notes provide an elementary introduction, within the framework of finite Quantum systems, to recent developments in the theory of entropic fluctuations.

  • entropic fluctuations in Statistical Mechanics i classical dynamical systems
    Nonlinearity, 2011
    Co-Authors: Vojkan Jaksic, Claudealain Pillet, Luc Reybellet
    Abstract:

    Within the abstract framework of dynamical system theory we describe a general approach to the transient (or Evans–Searles) and steady state (or Gallavotti–Cohen) fluctuation theorems of non-equilibrium Statistical Mechanics. Our main objective is to display the minimal, model independent mathematical structure at work behind fluctuation theorems. In addition to its conceptual simplicity, another advantage of our approach is its natural extension to Quantum Statistical Mechanics which will be presented in a companion paper. We shall discuss several examples including thermostated systems, open Hamiltonian systems, chaotic homeomorphisms of compact metric spaces and Anosov diffeomorphisms.

  • Topics in nonequilibrium Quantum Statistical Mechanics
    2006
    Co-Authors: Walter H. Aschbacher, Vojkan Jaksic, Yan Pautrat, Claudealain Pillet
    Abstract:

    These notes are an expanded and revised version of the lectures given by the second and fourth autor in the summer school "Open Quantum System" held in Grenoble, June 16-July 4, 2003. They provide an introduction to recent developments in non-equilibrium Statistical Mechanics of open Quantum systems, including a completely worked out (simple) example. We discuss non-equilibrium steady states (NESS) and their structural properties, entropy production, linear response theory and weak coupling limit. The emphasis is on Ruelle's scattering approach to the construction of NESS.

Vojkan Jaksic - One of the best experts on this subject based on the ideXlab platform.

  • a note on the landauer principle in Quantum Statistical Mechanics
    Journal of Mathematical Physics, 2014
    Co-Authors: Vojkan Jaksic, Claudealain Pillet
    Abstract:

    The Landauer principle asserts that the energy cost of erasure of one bit of information by the action of a thermal reservoir in equilibrium at temperature T is never less than kT log 2. We discuss Landauer's principle for Quantum Statistical models describing a finite level Quantum system S coupled to an infinitely extended thermal reservoir R. Using Araki's perturbation theory of KMS states and the Avron-Elgart adiabatic theorem we prove, under a natural ergodicity assumption on the joint system S+R, that Landauer's bound saturates for adiabatically switched interactions. The recent work of Reeb and Wolf on the subject is discussed and compared.

  • Quantum hypothesis testing and non equilibrium Statistical Mechanics
    Reviews in Mathematical Physics, 2012
    Co-Authors: Vojkan Jaksic, Yoshiko Ogata, Claudealain Pillet, Robert Seiringer
    Abstract:

    We extend the mathematical theory of Quantum hypothesis testing to the general W*-algebraic setting and explore its relation with recent developments in non-equilibrium Quantum Statistical Mechanics. In particular, we relate the large deviation principle for the full counting statistics of entropy flow to Quantum hypothesis testing of the arrow of time.

  • entropic fluctuations in Quantum Statistical Mechanics an introduction
    arXiv: Mathematical Physics, 2011
    Co-Authors: Vojkan Jaksic, Yoshiko Ogata, Yan Pautrat, Claudealain Pillet
    Abstract:

    These lecture notes provide an elementary introduction, within the framework of finite Quantum systems, to recent developments in the theory of entropic fluctuations.

  • entropic fluctuations in Statistical Mechanics i classical dynamical systems
    Nonlinearity, 2011
    Co-Authors: Vojkan Jaksic, Claudealain Pillet, Luc Reybellet
    Abstract:

    Within the abstract framework of dynamical system theory we describe a general approach to the transient (or Evans–Searles) and steady state (or Gallavotti–Cohen) fluctuation theorems of non-equilibrium Statistical Mechanics. Our main objective is to display the minimal, model independent mathematical structure at work behind fluctuation theorems. In addition to its conceptual simplicity, another advantage of our approach is its natural extension to Quantum Statistical Mechanics which will be presented in a companion paper. We shall discuss several examples including thermostated systems, open Hamiltonian systems, chaotic homeomorphisms of compact metric spaces and Anosov diffeomorphisms.

  • Topics in nonequilibrium Quantum Statistical Mechanics
    2006
    Co-Authors: Walter H. Aschbacher, Vojkan Jaksic, Yan Pautrat, Claudealain Pillet
    Abstract:

    These notes are an expanded and revised version of the lectures given by the second and fourth autor in the summer school "Open Quantum System" held in Grenoble, June 16-July 4, 2003. They provide an introduction to recent developments in non-equilibrium Statistical Mechanics of open Quantum systems, including a completely worked out (simple) example. We discuss non-equilibrium steady states (NESS) and their structural properties, entropy production, linear response theory and weak coupling limit. The emphasis is on Ruelle's scattering approach to the construction of NESS.