Classical statistical mechanics phase space

The set of microstates of a finite system in quantum statistical mechanics is a finite, discrete denumerable set of quantum states each characterized by an appropriate collection of quantum numbers. In classical statistical mechanics, the set of microstates fonn a continuous (and therefore infinite) set of points in f space (also called phase space). 

Is the temperature 1/0 related to the variance of the momentum distribution as in the classical equipartition theorem It happens that there is no simple generalization of the equipartition theorem of classical statistical mechanics. For the 2N dimensional phase space F = (xi. .. XN,pi,.. -Pn) the ensemble average for a harmonic system is... 

The starting point of classical statistical mechanics is the exact equation of evolution of the distribution function p in phase space the Liouville equation, which Prigogine always wrote in the form... 

Conclusion. In classical statistical mechanics the evolution of a many-body system is described as a stochastic process. It reduces to a Markov process if one assumes coarse-graining of the phase space (and the repeated randomness assumption). Quantum mechanics gives rise to an additional fine-graining. However, these grains are so much smaller that they do not affect the classical derivation of the stochastic behavior. These statements have not been proved mathematically, but it is better to say something that is true although not proved, than to prove something that is not true. 

In the general approach to classical statistical mechanics, each particle is considered to occupy a point in phase space, i.e., to have a definite position and momentum, at a given instant. The probability that the point corresponding to a particle will fall in any small volume of the phase space is taken proportional to die volume. The probability of a specific arrangement of points is proportional to the number of ways that the total ensemble of molecules could be permuted to achieve the arrangement. When this is done, and it is further required that the number of molecules and their total energy remain constant, one can obtain a description of the most probable distribution of the molecules in phase space. Tlie Maxwell-Boltzmann distribution law results. 

The form of the expressions in Eqs (5.98) and (5.114) is closely related to the classical expressions for the rate constant given in Section 5.1. The quantum mechanical trace becomes in classical statistical mechanics an integral over phase space [9] and the Heisenberg operators become the corresponding classical (time-dependent) functions of coordinates and momenta [8]. Thus, Eq. (5.78) is the classical version of Eq. (5.114). Furthermore, note that Eq. (5.98) is related to Eq. (5.49), i.e., the relevant classical (one-way) flux through Ro, at a given time, becomes S(R - Ro)(p/p)9(p/p), exactly as in Eq. (5.49). 

The term density matrix arises by analogy to classical statistical mechanics, where the state of a system consisting of N molecules moving in a real three-dimensional space is described by the density of points in a 6N-dimensional phase space, which includes three orthogonal spatial coordinates and three conjugate momenta for each of the N particles, thus giving a complete description of the system at a particular time. In principle, the density matrix for a spin system includes all the spins, as we have seen, and all the spatial coordinates as well. However, as we discuss subsequently we limit our treatment to spins. For simplicity we deal only with application to systems of spin % nuclei, but the formalism also applies to nuclei of higher spin. 

Classical statistical mechanics is concerned with the probability distribution of phase points. In a classical microcanonical ensemble the phase space density is constant. Loosely speaking, aU phase points with the same energy are equally likely. In consequence the number of states of the classical system in a given energy range E to E + dE is proportional to the volume of the phase space shell defined by this energy range. 

In parallel there exist some attempts trying to introduce a field theory (FT) starting from the standard description in terms of phase space [4—6], Of course, the best way to derive a FT for classical systems should consist in taking the classical limit of a QFT in the same way as the so called classical statistical mechanics is in fact the classical limit of a quantum approach. This limit is not so trivial and the Planck constant as well as the symmetry of wave functions survive in the classical domain (see for instance [7]). Here, we adopt a more pragmatic approach, assuming the existence of a FT we work in the spirit of QFT. 

In classical statistical mechanics, it follows from conservation of probability in phase space that... 

Under very general conditions, it follows from classical statistical mechanics that the equilibrium behavior of our fluid system is adequately described % the behavior of a Gibbskn ensemble of systems characterized by a canonical distribution (in energy) in phase space. This has two immediate consequences. First it specifies the spatial distribution of our N molecule system. The simultaneous probability that some first molecule center hes in the volume element dr whose center is at and etc., and the Nih molecule center lies in the volume element dr f whose center is at is... 

While elegant in its simplicity, such an approach is extremely problematic when computing properties of physical systems. To appreciate this, we need to introduce some fundamental relationships of classical statistical mechanics, The volume of phase space that can be accessed by the N hard disks in the example above is called the canonical partition function, Q ... 

In classical statistical mechanics, each particle is regarded as occupying a point in phase space, i.e. to have an exact position and momentum at any particular instant. The probability that this point will occupy any small volume of the phase space is taken to be proportional to the volume. The Maxwell-Boltzmann law gives the most probable distribution of the particles in phase space. 

Equation (10.24) is called the von Neumann equation after the mathematician John von Neumann, who originated the concept of the density matrix. It also is known as the Liouville equation because of its parallel to Liouville s classical statistical mechanical theorem on the density of dynamic variables in phase space. 

The phase space of interest refers to the particle s position in space and its momentum. As momentum can be described by classical statistical mechanics, equations of motion can be expressed in terms of the particle s position in space and its momentum, hence the term Phase Space Dynamics. Considering that an ion beam is composed of a large number of charged particles, it then follows that the optical properties of the beam can be described as a collection of such parameters. 

We consider only the equilibrium case so that the distribution of these points phase space is time-independent. In quantum statistical mechanics, we had a discrete list of possible states. In classical statistical mechanics, we have coordinates and momentum components that can range continuously. We denote the probability disttibution (probability density) for the ensemble by / and define the probability that the phase point of a randomly selected system of the ensemble will lie in the 6A -dimensional volume element d tNci pi to be... 

From elementary classical statistical mechanics for the canonical ensemble (constant NVT), we can relate the free energy G of any system to an integral of the Boltzmann factor over coordinate q) and momentum (p) phase spaces ... 

Let be a function of coordinates only, so a point in is specified by [3(A(ab - 1) - 1] coordinates and 3(A(ab - ) momenta. Identify the missing coordinate as the reaction coordinate s (so s becomes a coordinate normal to the hypersurface), and identify the momentum conjugate to s as p. Let C denote the [6(A(ab — 1) - 2]-dimensional hyperface in in which ps = 0. Assume that the % region of phase space is populated according to a Boltzmann equilibrium distribution then Liouville s theorem of classical statistical mechanics shows it will evolve into a Boltzmann equilibrium distribution at and hence also at C. Consider the one-way flux of this equilibrium ensemble of phase points through in the 5 —> P direction. This flux may be calculated quite generally, and using this calculation plus equation (2) yields... 

In the books of classical statistical mechanics, the properties of the ideal gas have been evaluated, so we can focus on the configurational contribution. With the Monte Carlo method [17-19], we explore the configuration phase subspace according to the canonical or grand canonical distributions. The corresponding trajectory is essentially the projection of the phase space onto the subspace of coordinates, becoming independent of the subspace of moments. 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

In classical molecular dynamics, on the other hand, particles move according to the laws of classical mechanics over a PES that has been empirically parameterized. By means of their kinetic energy they can overcome energetic barriers and visit a much more extended portion of phase space. Tools from statistical mechanics can, moreover, be used to determine thermodynamic (e.g. relative free energies) and dynamic properties of the system from its temporal evolution. The quality of the results is, however, limited to the accuracy and reliability of the (empirically) parameterized PES. 

We note that the integral over the energetically allowed phase space— that is, the classical level density (97)—was found in Fig. 20 to be in excellent agreement with the quantum-mechanical level density. This finding indicates that there is a valid correspondence between the quantum-mechanical two-state system and its classical mapping representation. A similar conclusion was drawn in a recent smdy of a mapped two-state problem, which focused on the Lyapunov exponents and the energy level statistics of the system [124, 235]. 

The important word in this sentence is predict. It is important, in my opinion, to make a distinction between existence and predictability. Prigogine himself said (much later, in La Fin des Certitudes, LG.7) Every dynamical system must, of course, follow a trajectory, solution of its equations, independently of the fact that we may or may not construct it. Thus, a trajectory exists but cannot be predicted. The impossibility of prediction is therefore related to the impossibility of defining an instantaneous state (in the framework of classical mechanics) as a limit of a finite region of phase space (thus a limit of a result of a set of measurements). For an unstable system, such a region will be deformed and will end up covering almost all of phase space. The necessity of introducing statistical methods appears to me to be due to the practical (rather than theoretical) impossibility of determining a mathematical point as an initial condition. 

Atomistic computer simulations are a statistical mechanical tool to sample configurations from the phase space of the physical system of interest. The system is uniquely treated by specifying the interactions between the particles (which are usually described as being pointlike), the masses of all the particles, and the boundary conditions. The interactions are calculated either on-the-fly by an electronic structure calculation (see Section 2.2.3) or from potential functions, which have been parametrized before the simulation by fitting to the results of electronic structure calculations or a set of experimental data. In the first case, one frequently speaks of AIMD (see Section 2.2.3), although the motion of the nuclei is still treated classically. 

The basic issue is at a higher level of generality than that of the particular mechanical assumptions (Newtonian, quantum-theoretical, etc.) concerning the system. For simplicity of exposition, we deal with the classical model of N similar molecules in a closed vessel "K, intermolecular forces being conservative, and container forces having a force-function usually involving the time. Such a system is Hamiltonian, and we assume that the potentials are such that its Hamiltonian function is bounded below. The statistics of the system are conveyed by a probability density function 3F defined over the phase space QN of our Hamiltonian system. Its time evolution is completely determined by Liouville s equation... 

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