lecture notes on statistical mechanics - Scott Pratt - Michigan State ...

Thus, the states are populated proportional to the factor e −βϵ i, which is the Boltzmann distribution,with β being identified as the inverse temperature. Again, the parameter λ is chosen to normalizethe probability. However, again the Lagrange multipliers for a given β only enforce the constraintthat the average energy is some constant, not the particular energy one might wish. Thus, one mustadjust β to find the desired energy, a sometimes time-consuming process.For any quantity which is conserved on the average, one need only add a corresponding Lagrangemultiplier. For instance, a multiplier α could be used to restrict the average particle number orcharge. The probability for being in state i would then be:p i = exp(−1 − λ − βϵ i − αQ i ). (1.9)Typically, the chemical potential µ is used to reference the multiplier,α = −µ/T. (1.10)The charge Q i could refer to the baryon number, electric charge, or any other conserved quantity.It could be either positive or negative. If there are many conserved charges, Q can be replaced by⃗Q and µ can be replaced by ⃗µ.Rather than enforcing the last Lagrange multiplier constraint, that derivatives w.r.t. the multiplierare zero, we are often happy with knowing the solution for a given temperature and chemicalpotential. Inverting the relation to find values of T and µ that yield specific values of the energyand particle number is often difficult, and as will be seen later on, it is often the temperature andchemical potentials that are effectively constrained in many physical situations.In most textbooks, the charge is replaced by a number N. This is fine if the number of particlesis conserved such as a gas of Argon atoms. However, the more general case includes cases of rathercomplicated chemical reactions, or multiple conserved charges. For instance, both electric chargeand baryon number are conserved in the hadronic medium in the interior of a star. Positrons andelectrons clearly contribute to the charge with opposite signs. Without apology, these <strong>notes</strong> willswitch between using N or Q depending on the context. As long as a sum is conserved, there is nodifference in referring to it as a number or as a charge.In statistical mechanics one first considers which quantities one wishes to fix, and which quantitiesone wishes to allow to vary (but with the constraint that the average is some value). This choicedefines the ensemble. The three most common ensembles are the micro-canonical, canonical andgrand-canonical. The ensembles differ by which quantities vary, as seen in Table 1. If a quantity isallowed to vary, then a Lagrange multiplier defines the average quantity for that ensemble.Ensemble Energy Chargesmicro-canonical fixed fixedcanonical varies fixedgrand canonical varies variesTable 1:Ensembles vary by what quantity is fixed and what varies.4