Introduction to statistical mechanics. The macroscopic and the microscopic states.

1. Introduction to statistical mechanics. • The macroscopic and the microscopic states. • Equilibrium and observation time. • Equilibrium and molecular motion. • Relaxation time. • Local equilibrium. • Phase space of a classical system. • Statistical ensemble. • Liouville’s theorem. • Density matrix in statistical mechanics and its properties. • Liouville’s-Neiman equation.

2. Introduction to statistical mechanics. From the seventeenth century onward it was realized that material systems could often be described by a small number of descriptive parameters that were related to one another in simple lawlike ways. These parameters referred to geometric, dynamical and thermal properties of matter. Typical of the laws was the ideal gas law that related product of pressure and volume of a gas to the temperature of the gas.

3. Microscopic and macroscopic states The main aim of this course is the investigation of general properties of the macroscopic systems with a large number of degrees of dynamically freedom (with N ~ 1020 particles for example). From the mechanical point of view, such systems are very complicated. But in the usual case only a few physical parameters, say temperature, the pressure and the density, are measured, by means of which the ’’state’’ of the system is specified. A state defined in this cruder manner is calleda macroscopic stateor thermodynamic state. On the other hand, from a dynamical point of view, each state of a system can be defined, at least in principle, as precisely as possible by specifying all of the dynamical variables of the system. Such a state is calleda microscopic state.

4. Averaging The physical quantities observed in the macroscopic state are the result of these variables averaging in the warrantable microscopic states. The statistical hypothesis about the microscopic state distribution is required for the correct averaging. To find the right method of averaging is the fundamental principle of the statistical method for investigation of macroscopic systems. The derivation of general physical lows from the experimental results without consideration of the atomic-molecular structure is the main principle of thermodynamic approach.

5. Zero Low of Thermodynamics One of the main significant points in thermodynamics (some times they call it the zero low of thermodynamics) is the conclusion that every enclosure (isolated from others) system in time come into the equilibrium state where all the physical parameters characterizing the system are not changing in time. The process of equilibrium setting is called the relaxation process of the system and the time of this process is the relaxation time. Equilibrium means that the separate parts of the system (subsystems) are also in the state of internal equilibrium (if one will isolate them nothing will happen with them). The are also in equilibrium with each other- no exchange by energy and particles between them.

6. Local equilibrium Local equilibriummeans that the system is consist from the subsystems, that by themselves are in the state of internal equilibrium but there is no any equilibrium between the subsystems. The number of macroscopic parameters is increasing with digression of the system from the total equilibrium

7. Classical phase system Let (q1, q2 ..... qs) be the generalized coordinates of a system with idegrees of freedom and (p1 p2..... ps) their conjugate moment. A microscopic state of the system is defined by specifying the values of (q1, q2 ..... qs, p1 p2..... ps). The 2s-dimensional space constructed from these 2s variables as the coordinates in thephase spaceof the system. Each point in the phase space(phase point)corresponds to a microscopic state. Therefore the microscopic states in classical statistical mechanics make a continuous set of points in phase space.

8. Phase Orbit If the Hamiltonian of the system is denoted by H(q,p),the motion of phase point can be along the phase orbit and is determined by the canonical equation of motion (i=1,2....s) (1.1) (1.2) Therefore the phase orbit must lie on a surface of constant energy (ergodic surface).

9.  - space and -space Let us define  - space as phase space of one particle (atom or molecule). The macrosystem phase space (-space) is equal to the sum of  - spaces. The set of possible microstates can be presented by continues set of phase points. Every point can move by itself along it’s own phase orbit. The overall picture of this movement possesses certain interesting features, which are best appreciated in terms of what we call a density function(q,p;t). This function is defined in such a way that at any time t, the number of representative points in the ’volume element’(d3Nq d3Np) around the point (q,p) of the phase space is given by the product (q,p;t) d3Nq d3Np. Clearly, the density function(q,p;t) symbolizes the manner in which the members of the ensemble are distributed over various possible microstates at various instants of time.

10. Function of Statistical Distribution Let us suppose that the probability of system detection in the volume ddpdqdp1.... dps dq1..... dqs near point (p,q) equal dw (p,q)= (q,p)d. The function of statistical distribution (density function) of the system over microstates in the case of nonequilibrium systems is also depends on time. The statistical average of a given dynamical physical quantity f(p,q) is equal: (1.3) The right ’’phase portrait’’ of the system can be described by the set of points distributed in phase space with the density . This number can be considered as the description of great (number of points) number of systems each of which has the same structure as the system under observation copies of such system at particular time, which are by themselves existing in admissible microstates

11. Statistical Ensemble The number of macroscopically identical systems distributed along admissible microstates with density defined as statistical ensemble. A statistical ensembles are defined and named by the distribution function which characterizes it. The statistical average value have the same meaning as the ensemble average value. An ensemble is said to be stationary if  does not depend explicitly on time, i.e. at all times (1.4) Clearly, for such an ensemble the average value<f> of any physical quantity f(p,q) will be independent of time. Naturally, then, a stationary ensemble qualifies to represent a system in equilibrium. To determine the circumstances under which Eq. (1.4) can hold, we have to make a rather study of the movement of the representative points in the phase space.

12. Lioville’s theorem and its consequences Consider an arbitrary "volume"  in the relevant region of the phase space and let the "surface” enclosing this volume increases with time is given by (1.5) where d(d3Nq d3Np). On the other hand, the net rate at which the representative points ‘’flow’’ out of the volume (across the bounding surface ) is given by (1.6) here v is the vector of the representative points in the region of the surface element d, while is the (outward) unit vector normal to this element. By the divergence theorem, (1.6) can be written as

13. (1.7) where the operation of divergence means the following: (1.8) In view of the fact that there are no "sources" or "sinks" in the phase space and hence the total number of representative points must be conserved, we have , by (1.5) and (1.7) (1.9) or (1.10)

14. The necessary and sufficient condition that the volume integral (1.10) vanish for arbitrary volumes  is that the integrated must vanish everywhere in the relevant region of the phase space. Thus, we must have (1.11) which is the equation of continuity for the swarm of the representative points. This equation means that ensemble of the phase points moving with time as a flow of liquid without sources or sinks. Combining (1.8) and (1.11), we obtain

15. (1.12) The last group of terms vanishes identically because the equation of motion, we have for all i, (1.13) From (1.12), taking into account (1.13) we can easily get the Liouville equation (1.14) where {,H} the Poisson bracket.

16. Further, since (qi,pi;t), the remaining terms in (1.12) may be combined to give the «total» time derivative of . Thus we finally have (1.15) Equation (1.15) embodies the so-called Liouville’s theorem. According to this theorem (q0,p0;t0)=(q,p;t) or for the equilibrium system (q0,p0)= (q,p),that means the distribution function is the integral of motion. One can formulate the Liouville’s theorem as a principle of phase volume maintenance.

17. Density matrix in statistical mechanics The microstates in quantum theory will be characterized by a (common) Hamiltonian, which may be denoted by the operator. At time t the physical state of the various systems will be characterized by the correspondent wave functions (ri,t), where the ri,denote the position coordinates relevant to the system under study. Let k(ri,t), denote the (normalized) wave function characterizing the physical state in which the k-th system of the ensemble happens to be at time t ; naturally, k=1,2....N. The time variation of the function k(t)will be determined by the Schredinger equation

18. (1.16) Introducing a complete set of orthonormal functionsn, the wave functions k(t)may be written as (1.17) (1.18) here,n*denotes the complex conjugate ofnwhileddenotes the volume element of the coordinate space of the given system. Obviously enough, the physical state of thek-thsystem can be described equally well in terms of the coefficients . The time variation of these coefficients will be given by

19. The physical significance of the coefficients is evident from eqn. (1.17). They are the probability amplitudes for the k-th system of the ensemble to be in the respective states n; to be practical the number represents the probability that a measurement at time t finds the k-th system of the ensemble to be in particular state n. Clearly, we must have (1.19) where (1.20)

20. clearly, the matrix element mn(t) is the ensemble average of the quantity am(t)an*(t) which, as a rule, varies from member to member in the ensemble. In particular, the diagonal element mn(t) is the ensemble average of the probability, the latter itself being a (quantum-mechanical) average. (for all k) (1.21) We now introduce the density operator as defined by the matrix elements (density matrix) (1.22)

21. Equation of Motion for the Density Matrix mn(t) Thus, we are concerned here with a double averaging process - once due to the probabilistic aspect of the wave functions and again due to the statistical aspect of the ensemble!! The quantity mn(t) now represents the probability that a system, chosen at random from the ensemble, at time t, is found to be in the state n. In view of (1.21) and (1.22) we have (1.23) Let us determine the equation of motion for the density matrixmn(t).

22. (1.24) Here, use has been made of the fact that, in view of the Hermitian character of the operator, H*nl=Hln. Using the commutator notation, Eq.(1.24) may be written as (1.25)

23. This equation Liouville-Neiman is the quantum-mechanical analogue of the classical equation Liouville. • Some properties of density matrix: • Density operator is Hermitian, +=- • The density operator is normalized • Diagonal elements of density matrix are non negative • Represent the probability of physical values