| What is Statistical Mechanics? | Classical microcanonical Ensemble: Teminologies and postulate | Classical Ideal Gas |
1. Thermodynamics: Equilibrium Thermodyanmics Non-equilibrium Thermodynamics (Transports) |
Empirical Laws. Describe macro systems with macro measurable quantities, e.g. energy, volume, no. of moles, pressure, etc. Central to Equilibrium Thermodynamics is the entropy function, S(U,V,N). If S(U,V,N) is known for the system then every macro properties of the system can be calculated (see Ch1 or Callen for review). Thermodyanmic theory DOES NOT provide method to calculate S from microscopic laws. Non-equilibrium thermodynamics assumes that the entropy function varies slowly with time during steady flow of current. Go to 3.. Molecular Dynamics |
| 2. (Equilibrium) Statistical Mechanics | So far, deals only with equilibrium properties. If the energy spectrum of the system is known, then all equilibrium properties can be calculated from one postulate: Postulate: For a closed system at equilibrium, all states with the same energy are equally probalble. |
| 3. Classical and Quantum Molecular Dynamics | Starts with micro-dyanmical laws (classical or quantum). Classical: integrate the equation of motion to obtain r(t) and p(t). Quantum: Solve for many-body wavefunction, psi(r,t). Calculate both equilibrium properties and dynamical properties by averaging over all the particles. In general, no analytical results. Current computer simulations can handle ~ million particles for classical dyanmics and ~ hundreds of particles for quantum dynamics. Analytical result exists for classical ideal gas (kinetic theory of gas). **The postulate of statistical mechanics is motivated from the classical ideal gas results.** |
On p.127, second paragraph, the first sentense is technically incorrect. What is worng with it?
There is NO essentiall difference in the formalism. The difference is in applications. Classical: Continuum energy spectrum and distinguishable particles. Use phase space rather than quantum state description. Quantum:Can have both contiuum and discrete energy spectrum, can have distinguisable and indistinguishable "particles". |
-> We only need ONE postualte for equilibrium statistical mechanics and it is stated for an isolated ("closed") system. For systems in contact with a reservoir (e.g. heat bath), then the system + reservoir can be regard as a large closed system. What is the practical meaning of a "closed" system? (Read 6.1) -> Key concept: The macroscopic conditions of a system are characterized by a few macro variables, e.g. for a homogeneous, single-component system, it is sufficient to specifiy the energy (E), the volume (V), and the number of particles or moles (N). On the other hand, there are many micro states which are consistent with one set of macro conditions. Classically, each micro state is represented by a point in the 6N phase space. However, mathematically a point has zero measure, therefore the proper counting of state is defined in terms of a density function times the volume element of phase space. (Eq. 6.3) -> There are two forms of the postulate: (1) In terms of density function. All micro states with energy between E and E + delta are equally probable (Eq. 6.7) From this, one can calculate all properties by performing a statistical average (Eq.6.8) This method is seldom in practice but is used in deriving general results. OR (2) Define the entropy function in terms of the number of allowable states. (Eq. 6.15) Calculate all other properties from the entropy function using thermodynamical relationships. One can show that (1) and (2) are equivalent using information theory (to be discussed later in the course). -> Allowable phase space. Gamma (E) = number of states with energy between E and E+ delta Sigma (E) = number of states with energy less than and equal to E. ** Note: As N goes to infifnity, Gamma(E) = Sigma(E+delta), for any delta, such that 0< delta/E < 1. See H.W. 1, problem 1. omega (E) = density of state = Gamma (E)/delta. It has an unit of 1/energy. (Personally, I don't like to define entropy in terms of omega, Eq. 6.28). -> Section 6.2 shows that the definition of entropy function (Eq. 6.15) possess all the properties of the entropy function as defined in thermodyanmics. See H.W. 1, problem 2. |
Classical Ideal Gas Model: (1) Point molecules (2) Non-interacting (3) Rigid container walls (V is fixed) (4) Distinguishable molecules (leads to Gibbs's paradox in Section 6.6) Calculation of allowable phase space: Hamiltonian: For a fixed E, it defines a sphere in 3N-dimension. Therefore, the volume of allowable phase space= V**N x volume of a 3N-sphere. (Eq. 6.45 and Eq. 6.52) Eq. 6.46 - Eq. 6.50 show you how to calculate the volume of a 3N-sphere. Entropy function for a classical ideal gas is given in Eq. (6.54) Other important mathematical identity: Stirling's formula: log n != n logn - n |