Grand canonical ensemble

1. Grand canonical ensemble The physical system represented by a grand canonical ensemble is in equilibrium with an external reservoir with respect to both matter and energy exchange. The chemical potential is introduced to specify the fluctuation of the number of particles. It is convenient to use the grand canonical ensemble when the number of particles of the system cannot be easily fixed. Especially in quantum systems, e.g., a collection of bosons or fermions, the number of particles is an intrinsic property (rather than an external parameter) of each quantum state. When we consider systems that can exchange particles and energy with a large reservoir, both  and T are dictated by the reservoir (they are the reservoir’s properties). In particular, the equilibrium is reached when the chemical potentials of a system and its environment become equal to one another. In equilibrium, there is no net mass transfer, though the number of particles in a system can fluctuate around its mean value (diffusive equilibrium). Reservoir UR, NR, T,  System E, N Quantum statistics (T, V, ): Gibbs factor Grand part. Func. Grand free energy

2. R S 2 1 Reservoir UR, NR, T,  System E, N 1 and 2 - two microstates of the system (characterized by the spectrum and the number of particles in each energy level) The changes U and N for the reservoir = -(U and N for the system) proportional to the probability that the system in the state  contains N particles and has energy E Gibbs factor the probability that the system is in state with energy E and N particles: the grand partition function or the Gibbs sum The index arefers to a specific microstate of the system, which is specified by the occupation numbersni{n1, n2,.....}. The summation consists of two parts: a sum over the particle number N and for each N, over all microscopic states i of a system with that number of particles. The systems in equilibrium with the reservoir that supplies both energy and particles constitute the grand canonical ensemble.

3. Systems with a fixed number of particles in contact with the reservoir, occupancy ni<<1 Systems which can exchange both energy and particles with a reservoir, arbitrary occupancy ni E4 E4 E3 E3 E2 E2 E1 E1 The energy is fluctuating, but the total number of particles is fixed. The role of the thermal reservoir is to fix the mean energy of each particle (i.e., each system). The identical systems in contact with the reservoir constitute the canonical ensemble. This approach works well for the high-temperature (classical) case, which corresponds to the occupation numbers≪1. When the occupation numbers are ~1, it is to our advantage to choose, instead of particles, a single quantum level as the system, with all particles that might occupy this state. Each energy level is considered as a sub-system in equilibrium with the reservoir, and each level is populated from a particle reservoir independently of the other levels. We will now consider a system of identical non-interacting particles at the temperature T. iis the energy of a single particle in the i state, ni is the occupation number (the occupancy) for this state:

4. The grand partition function: The Gibbs sum depends on the single-particle spectrum, the chemical potential, the temperature, and the occupancy. The latter depends on the nature of particles that compose a system (fermions or bosons). Thus, in order to treat the ideal gas of quantum particles, we will need the explicit formulae for m and ni for bosons and fermions. In thermodynamics: Grand free energy When the system is in thermodynamic equilibrium, Φ is a minimum. This can be seen by considering that dΦ is zero if the volume is fixed and the temperature and chemical potential have stopped evolving.

5. The Grand Partition Function of an Ideal Quantum Gas The sum is taken over all possible values of ni The partition function of each quantum level is independent of other levels. Each energy level is considered as a sub-system in equilibrium with the reservoir, and each level is populated from a particle reservoir independently of the other levels. Page 266: The “system” and the “reservoir” therefore occupy the same physical space. What is the mean occupancy of ith level? The probability of a state (ith level) to be occupied by ni quantum particles: