Particle systems have total energy and distribute energy among particles {statistical mechanics}|.
energy: particle
Particles have energy levels. Particles have possible energy levels. Particle energy level cannot be zero, because particles must move and so have kinetic energy. Particles always have at least minimum ground-state energy, because energy has quanta.
energy: distribution
Some particles have lower energy, and some have higher energy {distribution, energy}. Particles exchange energy by collisions or electronic transitions. Systems have average particle energy, which is higher for higher temperature and/or work. Large systems typically have only one particle-energy distribution, which has highest probability.
energy: total
Sum of particle energies equals total energy. Total energy equals average particle energy times particle number.
energy: types
Particles can have translational energy, vibrational energy, rotational energy, and electronic-transition energy, with different ground states and different quanta. At normal temperatures, vibrational energy is at ground state, electronic-transition energy is at ground state, and rotational energy is above ground state. Total energy distributes equally among possible translation, rotation, vibration, and electronic-transition energy levels, if there are pathways. Systems with large energy quanta have few particles at high-level energies. Systems with small energy quanta have more particles at high-level energies. Energy change does not change particle distribution much.
entropy
Energy distributions have entropy. Entropy change changes particle distribution. Systems with few particles or low temperatures have quantum states, easy transitions among states, and minimal entropy. Systems with many particles or high temperature have thermal states. Black-hole event horizons have random kinetic energy and cause thermal states. Thermal states have random kinetic energy and have maximum entropy.
entropy: degeneracy
Different particle-energy distributions can have same number of particles at each energy level. For example, if two same-type particles exchange energies, system has different particle-energy distribution, same total energy, and same number of particles at each energy level. Different particle-energy distributions with the same energy and same number of particles at each energy level make system phase. System has largest phase, which has highest probability and most even energy distribution possible at total energy. Largest phase has highest entropy.
state: fluctuation
If system is in largest phase, particles have lowest probability of returning to smaller regions, because largest phase has highest probability. If particle collision results in smaller phase, in shortest possible time, system returns to largest phase, because largest phase has highest probability. Hawking radiation requires large phase fluctuation.
For systems with many molecules at equilibrium at temperature, frequency distributions {Boltzmann distribution, statistics}| can plot frequency against molecule energy. y(E) = e^(- E / (k * T)), where E is molecule energy level, y(E) is frequency for molecule energy level, e is natural-logarithm base, k is Boltzmann constant, and T is absolute temperature.
energy
Particle-energy probability is partition number and is relative frequency of that energy in Boltzmann distribution. Most-probable energies are near average energy. Total energy is integral of Boltzmann distribution.
comparison
At temperatures above 50 K, Boltzmann distributions look like Gaussian distributions.
equilibrium
Systems at equilibrium have Boltzmann distribution, because that distribution has much higher probability than other distributions with same total energy. Boltzmann distribution has the most combinations that can give total energy.
equilibrium: entropy
For that reason, Boltzmann distribution has lowest probability of molecule being in any one energy level, so Boltzmann distribution has the most entropy and least order. Entropy S equals Boltzmann constant k times combination-number C natural logarithm: S = k * ln(C).
Molecular properties {canonical properties} can be at constant temperature {canonical property}, at constant temperature and volume {grand canonical property}, or in isolated adiabatic systems {microcanonical property}.
If system molecules are indistinguishable, some particle-energy distributions have same numbers of particles at each energy level {degeneracy, system}|.
Particles have different possible motions and kinetic energies {degrees of freedom, energy}.
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.
Complex systems can have sets {ensemble, system}| of identical objects. Sets have statistical properties. Linear ensemble operators can calculate set-property average values, while varying initial conditions.
Collisions interchange energy, so average energies of system kinetic-energy sources are equal {equipartition, statistical mechanics}|.
motions
For particles, kinetic energy partitions equally into available motion states {degrees of freedom, motion}. Particles have different possible motions. Translations can independently be in three spatial dimensions. Vibrations depend on chemical-bond stretching-and-bending modes. Rotations can be in zero to three rotation dimensions, depending on molecule symmetry.
energy
If temperature is above 50 K, average partition energy E is half Boltzmann constant k times absolute temperature T: E = 0.5 * k * T. System partition function is product of particle partition functions.
In rare systems, all phases solve energy equation, and system reaches equilibrium {ergodic hypothesis}|. Though few systems are ergodic, real systems come arbitrarily close to ergodic {quasiergodic hypothesis}. Quasiergodic systems are fractal, because one trajectory cannot fill up space but can pass close to all points. If trajectory does not follow simple law, system uses statistical law.
Plane rectangles can have many square cells and some particles {gas in box model}. Connected cells make a region with percentage of total cell number. For example, box can have 10 cells, with one-cell region in one corner. Probability that one particle is in region is 1/10.
Statistical thermodynamics applies to systems {ideal fluid} in which the only interactions among particles are elastic collisions, with no forces between molecules. Particles can have cross-sections.
Boltzmann distribution gives number of molecules at each energy level {partition function}| {canonical partition function}, for a temperature.
Probability {partition number, energy} of particle energy level is relative frequency of that energy in Boltzmann distribution. Most-probable energies are near average energy.
Molecule collisions make fast and discrete molecule-energy changes {quantum energy change}.
temperature
Energy fluctuations Q depend on Boltzmann constant k times absolute temperature T: Q = k*T. At higher temperatures, energy change is more, and molecule energy levels are farther apart. At low temperature, quanta are almost equal, and molecules have ground-state energy Q0 plus multiple of energy quantum Q: Q0, Q0 + 1*Q, Q0 + 2*Q, Q0 + 3*Q, and so on.
factors
Quantum increases if volume decreases, system does work, mass decreases, temperature decreases, pressure decreases, fields decrease, or electrons transition to lower orbits. In those cases, overall energy decreases, so quanta are bigger.
entropy
If system has only entropy changes, quanta stay the same.
high energy
If quanta are large, high energies are hard to reach.
Spontaneous processes {spontaneous process}| lower free energy. Particles move along geodesics. Electrons move along zero-field lines. Particles orbit at lowest orbit.
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Date Modified: 2022.0225