Statistical Physics

Full Title

Complexity does not permit the exact determination of all states for even so much as a single atom larger than one proton and one elsectron.

Context

• From the time of Gauss probability was used in experimental physics to estimate error bound in experiments. But the assumption was that the underlying physical laws were absolute and admitted to no variability.
• Bernoulli, Clausius and Maxwell developed the idea that motions of atoms in a gas could be used to explain heat and the idea of entropy which can never decrease in closed systems.

Boltzmann

The starting idea for classical statistics is the Hamiltonian phase space of 6N dimensions where N is the number of particles. So each particle has its own unique 3 position and 3 momentum values which are tracked over time. See the wiki page on Identical Particle for details on this concept.

Boltzmann's statistics formula for Entropy:

SN = NkB ln(W)

Planck

From the beginning of Boltzmann's statistical approach Planck, among many others, rejected the Boltzmann's statistical approach. But, in the end, both Boltzmann and Planck had to accept what the math was telling them. Some time any complex physical system will do unexpected things. This was a an unwelcome outcome for logical minds. This rear-guard action against any statistical solution continues even to this day. Almost no one seems to like the possibility that everything that happens, from the very lowest event, is up to chance.

When Planck tried to solve the problem of black-body radiation, where the then current theories predicted that the amount of radiation should continuously increase as the frequency increased, his result looked a lot like Boltzmann's solution for the distribution of heat in an ideal gas. This was not the solution that Plank expected, but he finally had to accept that a statistical solution was required, but with a difference. In Planck's solution he took the unexpected step of treating the quanta of energy as indistinguishable from every other quanta of energy with the same energy level. When he did that he arrived at his famous formula E = hν.

In the years leading up to 1900 Planck found this formula which states that the density of cavity radiation is directly related to the average internal energy of the oscillators using classical physics.

ρ(ν,T) = 8πν2 U / c3

From this Planck was able to derive the statistics formula for entropy

SN = NkB ln( (1+U/hν)exp(1+U/hν) / (U/hν)exp(U/hν) )

Plank then derived the entropy from statistics to get the following formula,

SN = NkB ln( (1+U/E)exp(1+U/E) / (U/E)exp(U/E) )

Comparing these two gives Planck's most famous formula

E = hν

Bose-Einstein

S. N. Bose read Planck's papers and determined that quantum statistics should not depend on classical physics the way Planck did. The following equation just asserts that the total energy is just equal to the sum of energies of every particle in the system of interest (aka the ensemble).

E = Σ Ns hνs. Sum over all states.

The values N_s will exactly determine the total energy in the same manner as Feynman was able to use the sum of all paths (see below) to determine the total probably of each possible event for a particle in motion.

Wave number, a unit of frequency, often used in atomic, molecular, and nuclear spectroscopy, equal to the true frequency divided by the speed of the wave and thus equal to the number of waves in a unit distance. In the case of light, the frequency, symbolized by the Greek letter nu (ν), of any wave equals the speed of light, c, divided by the wavelength λ: thus ν = c/λ.

Consequences

Statistical analysis become necessary when dealing with particles that were too numerous to track individually. It turns out the the very success of Newtonian physics and the calculus was to assume that differences could be made small enough that calculations could be easily performed. When the reality of particles that were not continuous became part of the study of physics, then Statistical Physics became necessary to understand physical reality. We find that in Quantum Mechanics probability and statistics becomes the only way to make any sort of predictions at all. Richard Feynman that his to say about the uncertainty implicit in a statistical approach. "The uncertainty principle “protects” quantum mechanics. Heisenberg recognized that if it were possible to measure the momentum and the position simultaneously with a greater accuracy, the quantum mechanics would collapse. So he proposed that it must be impossible. Then people sat down and tried to figure out ways of doing it, and nobody could figure out a way to measure the position and the momentum of anything—a screen, an electron, a billiard ball, anything—with any greater accuracy. Quantum mechanics maintains its perilous but still accurate mexistence."