Boltzmann Distribution and Helmholtz Free Energy The Boltzmann

Boltzmann Distribution and Helmholtz Free Energy The Boltzmann factor - We consider a thermodynamic system (S) in contact with a Reservoir (R) (Heat Bath), which has temperature T. S+R is an isolated system with U 0 internal energy. We assume S<<R - central problem: what is the probability that S will be is a given state : s (S is not an isolated system, and we cannot use the Ps=1/g results)? - Since S+R is an isolated system, and thus we can apply our fundamental assumption for the probability of the states. - When S is in state s with energy s, R will have an energy UR=U 0 - s with multiplicity g. R(U 0 - s ). - applying our fundamental assumption for the total isolated system and the relation between entropy and multiplicity, for two states (1 and 2) with energies 1 and 2 we get:

We assume now i<<U 0, and write Boltzmann factor and we get the important relation: partition function: - The partition function is the proportionality factor between the probability P( s) and Boltzmann factor! Normalization satisfied! One of the MOST USEFUL and IMPORTANT results of statistical physics Average energy of the S thermodynamic system <…> --> ensemble average (sometimes called thermal average)

Energy and heat capacity of a two state system We consider a thermodynamic system with two possible states (energies 0 and ) in contact with a heath-bath at temperature T. We calculate the internal energy and heat capacity of the system as a function of the system temperature. 0 for >> for <<

Pressure We consider a system in a quantum state s of energy s. We assume that s depends on volume V (Example quantum particle closed in a 3 d box) We consider a quasi-static volume change: V--> V-d. V calculation of pressure (during the quasi-static and reversible process the total number of micro-states were kept constant--> entropy was kept constant) for “isentropic” process:

Thermodynamic identity The combined mathematical form of the first and second law of thermodynamics (when the particle number is fixed) Problems: For a binary model system formed by 4 spins with magnetic moments +/- m in magnetic field B and in contact with a reservoir at temperature T find: 1. The probability that the total magnetization is 4 m. 2. The probability that the total magnetization is 0. At which temperature will this probability become 1? 3. The probability that one randomly selected spins is pointing in the + direction. 4. The average energy of the system. Extra problem: Consider a binary model system formed by 4 spins with magnetic moments +/- m in magnetic field B and in contact with a reservoir at temperature T. By using the Renyi entropy formula (see the extra problem from the end of lecture-note 3) , calculate the entropy at an arbitrary temperature T. Generalize the result for N spins.