Difference between revisions of "Statistical Physics"

Revision as of 11:57, 15 June 2023

Full Title

Complexity

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. Bot, in the end, not Boltzmann and Planck had to accept what the math was telling them. Some time the system will do unexpected things. This was a an unwelcome outcome for logical minds.^[1] 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.

Planck's statistics formula for entropy

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

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 because part of the study of physics, then Statistical Physics became necessary. 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 correct existence."^[2]

References

1. ↑ Jim Baggot Quantum Cook Cookbook Oxford (2020) ISBN = 9780198827863
2. ↑ Richard Feynman Quantum Behavior (1965) https://www.feynmanlectures.caltech.edu/III_01.html