The fundamentals of Statistical Mechanics require a fresh definition in the context of the developments in Classical Mechanics of integrable and chaotic systems. This is done with the introduction of Micro Partitions ; a union of disjoint components in phase space. Partition functions including the invariants, Kolmogorov Entropy and Euler number are introduced. The ergodic hypothesis for partial ergodicity is discussed. In the context of Quantum Mechanics the presence of symmetry groups with irreducible representations gives rise to degenerate and non degenerate spectrum for the Hamiltonian. Quantum Statistical Mechanics is formulated including these two cases ; by including the multiplicity dimension of the group representation and the Casimir invariants into the Partition function. The possibility of new kinds of phase transitions is discussed. The occurence of systems with non simply connected configuration spaces and Quantum Mechanics for them, also requires a possible generalisation of Statistical Mechanics. The developments of Quantum pure, mixed, and entangled states has made it neccessary to understand the Statistical Mechanics of the multipartite N particle system. And to obtain in terms of the density matrices, written in energy basis, the Trace of the Gibbs operator as the Partition function and use it to get statistical averages of operators. There are some issues of definition, interpretation and application that are discussed.
The fundamentals of Statistical Mechanics require a fresh definition in the context of the developments in Classical Mechanics of integrable and chaotic systems. This is done with the introduction of Micro Partitions ; a union of disjoint components in phase space. Partition functions including the invariants, Kolmogorov Entropy and Euler number are introduced. The ergodic hypothesis for partial ergodicity is discussed. In the context of Quantum Mechanics the presence of symmetry groups with irreducible representations gives rise to degenerate and non degenerate spectrum for the Hamiltonian. Quantum Statistical Mechanics is formulated including these two cases ; by including the multiplicity dimension of the group representation and the Casimir invariants into the Partition function. The possibility of new kinds of phase transitions is discussed. The occurence of systems with non simply connected configuration spaces and Quantum Mechanics for them, also requires a possible generalisation of Statistical Mechanics. The developments of Quantum pure, mixed, and entangled states has made it neccessary to understand the Statistical Mechanics of the multipartite N particle system. And to obtain in terms of the density matrices, written in energy basis, the Trace of the Gibbs operator as the Partition function and use it to get statistical averages of operators. There are some issues of definition, interpretation and application that are discussed.
Thermodynamics is used to describe the bulk properties of matter in or near equilibrium. Many scientists, notably Boyle, Carnot, Clausius, Gay-Lussac, Gibbs, Joule, Kelvin, and Rumford, contributed to the development of the field over three centuries. Quantities such as mass, pressure, energy, and so on are readily defined and measured. Classical statistical mechanics attempts to understand thermodynamics by the application of classical mechanics to the microscopic particles making up the system. Great progress in this field was made by physicists like Boltzmann and Maxwell. Temperature, entropy, particle number, and chemical potential are thus understandable in terms of the microscopic nature of matter. Classical mechanics is inadequate in many circumstances however, and ultimately must be replaced by quantum mechanics. In fact, the ultraviolet catastrophe encountered by the application of classical mechanics and electromagnetism to blackbody radiation was one of the problems that led to the development of quantum theory. The development of quantum statistical mechanics was achieved by a number of twentieth century physicists, most notably Planck, Einstein, Fermi, and Bose. The purpose of this chapter is to give a mini-review of the basic concepts of quantum statistical mechanics as applied to noninteracting systems of particles. This will set the stage for the functional integral representation of the partition function, which is a cornerstone of modern relativistic quantum field theory and the quantum statistical mechanics of interacting particles and fields. Ensembles One normally encounters three types of ensemble in equilibrium statistical mechanics. The microcanonical ensemble is used to describe an isolated system that has a fixed energy E , a fixed particle number N , and a fixed volume V .
Review of quantum statistical mechanicsThermodynamics is used to describe the bulk properties of matter in or near equilibrium.Many scientists, notably Boyle, Carnot, Clausius, Gay-Lussac, Gibbs, Joule, Kelvin, and Rumford, contributed to the development of the field over three centuries.Quantities such as mass, pressure, energy, and so on are readily defined and measured.Classical statistical mechanics attempts to understand thermodynamics by the application of classical mechanics to the microscopic particles making up the system.Great progress in this field was made by physicists like Boltzmann and Maxwell.Temperature, entropy, particle number, and chemical potential are thus understandable in terms of the microscopic nature of matter.Classical mechanics is inadequate in many circumstances however, and ultimately must be replaced by quantum mechanics.In fact, the ultraviolet catastrophe encountered by the application of classical mechanics and electromagnetism to blackbody radiation was one of the problems that led to the development of quantum theory.The development of quantum statistical mechanics was achieved by a number of twentieth century physicists, most notably Planck, Einstein, Fermi, and Bose.The purpose of this chapter is to give a mini-review of the basic concepts of quantum statistical mechanics as applied to noninteracting systems of particles.This will set the stage for the functional integral representation of the partition function, which is a cornerstone of modern relativistic quantum field theory and the quantum statistical mechanics of interacting particles and fields. EnsemblesOne normally encounters three types of ensemble in equilibrium statistical mechanics.The microcanonical ensemble is used to describe an isolated system that has a fixed energy E, a fixed particle number N , and a fixed 1
The exchange of conserved variables between a subsystem and a reservoir or environment entangles their wave functions. Their subsequent collapse, together with the uniform weight of wave space, gives rise to quantum statistical mechanics formulated as the trace of the probability and observable operators. A dual representation in position and momentum space allows a permutation loop expansion for particle symmetrization, and a second expansion accounting for the non-commutativity of the position and momentum operators. The leading term gives classical statistical mechanics, and higher-order terms give successive quantum corrections that become increasingly important for low temperatures, high densities and small masses. The probability density in phase space is stable under the stochastic dissipative equations of motion. Trapped Bose-Einstein condensates may be modeled as a system of non-interacting harmonic oscillators. Most bosons in the condensate are in the ground state, in contrast to the -transition in 4He where the fraction in the ground state is negligible.
We have thus far looked at the classical statistical mechanics of a collection of Ising spins in terms of their partition function. We have seen how to extract the free energy per site and correlation functions of the d = 1 Ising model. We will now consider a second way to do this. While the results are, of course, going to be the same, the method exposes a deep connection between classical statistical mechanics in d = 1 and the quantum mechanics of a single spin-12 particle. This is the simplest way to learn about a connection that has a much wider range of applicability: the classical statistical mechanics of a problem in d dimensions may be mapped on to a quantum problem in d – 1 dimensions. The nature of the quantum variable will depend on the nature of the classical variable. In general, the allowed values of the classical variable ( s =±1 in our example) will correspond to the maximal set of simultaneous eigenvalues of the operators in the quantum problem ( σz in our example). Subsequently we will study a problem where the classical variable called x lies in the range –∞ < x < ∞ and will correspond to the eigenvalues of the familiar position operator X that appears in the quantum problem. The correlation functions will become the expectation values in the ground state of a certain transfer matrix . The different sites in the d = 1 lattice will correspond to different, discrete, times in the life of the quantum degree of freedom. By the same token, a quantum problem may be mapped into a statistical mechanics problem in one-higher dimensions, essentially by running the derivation backwards. In this case the resulting partition function is also called the path integral for the quantum problem. There is, however, no guarantee that the classical partition function generated by a legitimate quantum problem (with a Hermitian Hamiltonian) will always make physical sense: the corresponding Boltzmann weights may be negative, or even complex! In the quantum mechanics we first learn, time t is a real parameter. For our purposes we also need to get familiar with quantum mechanics in which time takes on purely imaginary values t = –iτ , where τ is real. It turns out that it is possible to define such a Euclidean quantum mechanics .
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.
Abstract The functional integral representation of the density matrix at thermal equilibrium in non-relativistic quantum mechanics (QM) with many degrees of freedom, in the grand canonical formulation is introduced. In QM, Hamiltonians H(p,q) can be also expressed in terms of creation and annihilation operators, a method adapted to the study of perturbed harmonic oscillators. In the holomorphic formalism, quantum operators act by multiplication and differentiation on a vector space of analytic functions. Alternatively, they can also be represented by kernels, functions of complex variables that correspond in the classical limit to a complex parametrization of phase space. The formalism is adapted to the description of many-body boson systems. To this formalism corresponds a path integral representation of the density matrix at thermal equilibrium, where paths belong to complex spaces, instead of the more usual position–momentum phase space. A parallel formalism can be set up to describe systems with many fermion degrees of freedom, with Grassmann variables replacing complex variables. Both formalisms can be generalized to quantum gases of Bose and Fermi particles in the grand canonical formulation. Field integral representations of the corresponding quantum partition functions are derived.