Physical systems can have different numbers and energy levels of particles {distribution of energies} {energy distribution}. Particles can be molecules, atoms, photons, or subatomic particles.
energy quanta
Particle energy cannot be zero, because particles are always moving and so have kinetic energy. Particle energy has quanta, by quantum mechanics, so particles have lowest energy level {ground-state energy}. Particle energies increase from ground-state energy by discrete energy quanta. Possible particle energies are ground-state energy, ground-state energy plus one quantum, ground-state energy plus two quanta, and so on. For total energy, possible energy levels have numbers of particles. Systems have particles at ground-state energy, particles at ground-state energy plus one quantum, particles at ground-state energy plus two quanta, and so on. Particle number at high energy levels is small compared to number at low energy levels, because elastic collisions distribute energy among energy levels. High particle energy has low probability. Infinite particle energy has zero probability.
system energy
Closed systems have constant total energy. Total energy is ground-state energy times particle number, plus any quanta. Sum of particle energies makes total energy. Product of particle number and ground-state energy is minimum system energy.
energy distribution
For example, two-particle system can have one particle with energy 3, one particle with energy 1, and total energy 4. For closed systems, particle collisions can change energy distribution, but total energy stays constant. For example, the two-particle system can have one particle with energy 2, one particle with energy 2, and total energy 4.
energy distribution: low-energy example
Two-particle system can have ground-state energy Q0, one particle at ground-state energy, E1 = Q0, and another particle at one quantum energy level Q above ground-state energy, E2 = Q0 + 1*Q. Total energy is E1 + E2 = Q0 + (Q0 + 1*Q) = 2*Q0 + 1*Q. See Figure 1.
energy distribution: equivalent distributions
For closed systems, different energy distributions can result in same total energy. For example, twelve-molecule systems can have energy distributions in which each particle has energy Q1a and total energy is 12*Q1a. By particle collision, system can have energy distribution with six molecules one quantum Q above Q1a and six molecules one quantum Q below Q1a. System still has total energy 12*Q1a.
For two-molecule system with total energy 2*Q0 + 2*Q, both molecules can have energy Q0 + 1*Q. After collisions, first molecule can have energy Q0, and second molecule can have energy Q0 + 2*Q, or first molecule can have energy Q0 + 2*Q, and second molecule can have energy Q0. See Figure 2. All three energy distributions have same total energy.
probability
In closed physical system, all energy distributions have same total energy, and all distributions are equally likely, because collisions transfer energy freely between particles.
probability: distinguishable particles
If particles are distinguishable, energy distributions are unique and have equal probability. For example, system can have total energy 6, ground-state energy 2, quantum 1, and 2 particles. If particles are distinguishable, such as E1 and E2, three energy distributions are possible. E1 = 2 and E2 = 4. E1 = 4 and E2 = 2. E1 = 3 and E2 = 3. Distribution E1 = 3 and E2 = 3 has one-third probability. Distribution E1 = 2 and E2 = 4 has one-third probability. Distribution E1 = 4 and E2 = 2 has one-third probability. Distributions are equally likely. See Figure 3.
In this system, particles cannot have energy 0 or 1, because ground-state energy is 2. Only these three cases make total energy 6.
probability: indistinguishable particles
Typically, some system particles are exactly the same and so indistinguishable. For example, all electrons are the same. If particles are indistinguishable, some energy distributions appear the same.
For example, system can have total energy 6, ground-state energy 2, quantum 1, and 2 indistinguishable particles. Two energy distributions are possible: energy 2 and energy 4 or energy 3 and energy 3. Cases E1 = 2 and E2 = 4, and E1 = 4 and E2 = 2, are now indistinguishable. Energy distribution 3 and 3 happens once and has one-third probability. Energy distribution 2 and 4 happens twice and has two-thirds probability.
probability: degeneracy
If particles are indistinguishable, some energy distributions have same numbers of particles at each energy level. In the example, two energy distributions have one particle at level 2 and one particle at level 4, so degeneracy is two.
Degenerate energy-distribution probability is degeneracy divided by number of distributions when particles are distinguishable. In the example, number of energy distributions with distinguishable particles is three. Degeneracy of "energy 2/energy 4" distribution is two, and probability is 2/3. Degeneracy of "energy 3/energy 3" distribution is one, and probability is 1/3.
For degenerate distributions, more degeneracy makes higher probability. Degeneracy is greater if most particles are near average energy and particles have Boltzmann energy distribution. Maximum degeneracy spreads particles maximally. For many-particle systems, highest probability is many orders of magnitude above second-most-likely distribution.
partition number
If system has constant total energy, distribution degeneracy {partition number, distribution} is (total number of particles)! / (number at ground-state energy)! * (number at ground-state energy plus one quantum)! * ... * (number at ground-state energy plus infinite number of quanta)!, where ! means factorial. Above 50 K, for thermal distributions, partition number maximizes according to Boltzmann distribution.
probability: particle
Particle has probability that it is in energy level. If system is in energy distribution with maximum degeneracy, particle has lowest average probability that it is in energy level, because particles spread most evenly. Other energy distributions increase average probability that particle is in energy level, because they concentrate particles more.
system state
With collisions, systems tend to go to most degenerate energy distribution, from which the most collisions make same degenerate distribution, because it repeats itself the most. This is why the most-degenerate energy distribution has highest probability. Isolated systems soon reach this single stable state and stay there.
Physical Sciences>Physics>Heat>Statistical Mechanics
5-Physics-Heat-Statistical Mechanics
Outline of Knowledge Database Home Page
Description of Outline of Knowledge Database
Date Modified: 2022.0224