Entropy and temperature Fundamental assumption : an isolated system (N, V and U and all external parameters constant) is equally likely to be in any of the quantum states accessible to it! A quantum state is accessible if its properties are compatible with the physical specification of the system Probability of a state s: P(s). Let g be the number of accessible states --> P(s)=1/g if the state s is accessible, and P(s)=0 otherwise Properties: Ensemble: a collection of many “copies” (replica) of the original system. If there are g accessible states there are g replicas, one in each accessible quantum state. Averages calculated by us are all ensemble averages Ergodic hypothesis: the thermodynamic systems goes over all accessible states in a very short time (in a time much shorter than needed for the measurement of a macroscopic parameter) Basic hypothesis for an isolated system: time average equals the ensemble average Example: Ensemble of a system of 10 spins with energy -8mB and spin exces 2s=8.

Most probable configuration: - Let two systems (S1 and S2 be in contact) so that energy can be transferred from one to the other (thermal contact) - S=S1+S2 another larger system. We have for internal energy: U=U 1 +U 2. - let us consider S isolated (U, V, N….constant) - what determines whether there will be a net flow of energy from one system to another? --> concept of temperature - the most probable division of the total energy is that for which the combined system has the maximum number of accessible states! - let us consider the case of thermal contact of two systems. The numbers of particles in system 1 is N 1 and in system 2 is N 2, the values of the energies in system 1 is U 1 and in system 2 U 2. The total energy U=U 1 +U 2 is fixed. (g(N,U) is the multiplicity function of system S, g 1,2 for system 1,2 respectively.) a configuration : set of all states with specified values of U 1 and U 2.. most probable configuration: the configuration for which g 1 g 2 is maximum - if N>>1 fluctuations about the most probable configurations are small Schematic representation of the dependence of the configuration multiplicity on the division of the total energy between the systems S1 and S2 - values of the average physical properties of a large system in thermal contact with another large system are accurately described by the most probable configuration! --> such average values are called thermal equilibrium values average of a physical quantity over all accessible states --> average over the most probable configuration

let us consider the case of thermal contact of two spin systems in magnetic field. The numbers of spins in system 1 is N 1 and in system 2 is N 2, the values of the spin excess 2s 1, 2s 2 may be different and can change but the total energy: U=U 1 (s 1 )+U 2 (s 2 )=-2mB(s 1 +s 2 )=-2mBs is fixed. (g(N,s) is the multiplicity function of system S, g 1,2 for system 1,2 respectively. If N 1 <N 2 then in the sum -1/2N 1  s 1  1/2N 1 ) Example: Two spin systems in thermal contact the most probable configuration of the system is where: Nearly all accessible states of the combined system satisfy or nearly satisfy (*)! (*) sharpness of g 1 g 2: Numerical example: N 1 =N 2 =10 22;  =10 12 ;  /N 1 = > 2  2 /N 1 =200 --> g 1 g 2 reduced to e -400  of its maximal value! We may expect to observe substantial fractional deviations from the most probable configuration only for a small system in thermal contact with a large system!

Thermal equilibrium consider the case of thermal contact of two systems. The numbers of particles in system 1 is N 1 and in system 2 is N 2, the values of the energies in system 1 is U 1 and in system 2 U 2. The total energy U=U 1 +U 2 is fixed. (g(N,U) is the multiplicity function of system S, g 1,2 for system 1,2 respectively.) in thermal equilibrium the largest term in the above sum governs the properties of the total system! in thermal equilibrium: we define a quantity , called entropy In thermal equilibrium we have:

The concept of temperature: - in thermal equilibrium the temperatures of the two systems are equal! one can define the absolute temperature as: k B is the Boltzmann constant, k B =1.381 x J/K the fundamental temperature,  :  has the dimensions of energy Temperature scales: (defined experimentally by thermometers ):

Problems: 1. Solve problem 1. (“Entropy and temperature” on page 52) 2. Solve problem 2. (“Paramagnetism” on page 52) 3. Calculate the multiplicity function for a harmonic oscillator (example on page 24) 4. Solve problem 3. (“Quantum harmonic oscillator” on page 52) Extra problem: Consider a binary magnetic system formed by 5 spins ( ) arranged in a row. The spins are interacting only between themselves, each spin with it’s two (or one - for spin 1 and 5) nearest neighbor. The interaction energy between two nearest neighbor spin i and k is: U(i,k)=-  i  k.. Let U be the total energy of the system. Determine the g(U) multiplicity function for this system. (Write up the possible values for U, and correspond g values)