Astroparticle Physics

390 B Results from Statistical Physics: Thermodynamics of the Early UniverseB.1 Statistical Mechanics Review“The general connection between energyand temperature may only be establishedby probability considerations.Two systems are in statistical equilibriumwhen a transfer of energy does notincrease the probability.”Max Planckfundamental postulate:equipartition of energyequilibrium distributionN-particle wave functionConsider a system with volume V = L 3 and energy U,which could be a cube of the very early universe. The numberof particles will not be fixed since the temperatures consideredhere will be so high that particles can be continuallycreated and destroyed. For the moment only a single particletype will be considered but eventually the situation will begeneralized to include all possible types.The system can be in any one of a very large numberof possible microstates. The fundamental postulate of statisticalmechanics is that all microstates consistent with theimposed constraints (volume, energy) are equally likely. Agiven microstate, e.g., an N-particle wave function ψ(x 1 ,...,x N ) specifies everything about the system, but this isfar more than one wants to know. To reduce the informationto a more digestible level, one can determine from the microstatethe momentum distribution of the particles, i.e., theexpected number of particles in each cell d 3 p of momentumspace.There will be many microstates that lead to the same distribution,but one distribution in particular will have overwhelminglymore possible microstates than the others. Togood approximation all the others can be ignored and thisequilibrium distribution can be regarded as the most likely.Once it has been found, one can determine from it the otherquantities needed, such as the energy density and pressure.So, to find the equilibrium distribution one needs to determinethe number of possible microstates consistent witha distribution and then find the one for which this is a maximum.This is treated in standard books on statistical mechanics,e.g., [47]. Here only the main steps will be reviewed.It is assumed that the system’s N-particle wave functioncan be expressed as a sum of N terms, each of which is theproduct of N one-particle wave functions of the formψ A (x) ∼ e ip A·x .(B.2)