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# Statistical physics

## Statistical mechanics

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### Kinetic theory of gases

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Theory that gases are made up of a bunch of small billiard balls that don't interact with each other.
This theory attempts to deduce/explain properties of matter such as the equation of state in terms of classical mechanics.

## Maxwell-Boltzmann vs Bose-Einstein vs Fermi-Diract statisics

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A good conceptual starting point is to like the example that is mentioned at The Harvest of a Century by Siegmund Brandt (2008).
Consider a system with 2 particles and 3 states. Remember that:
Therefore, all the possible way to put those two particles in three states are for:
• Maxwell-Boltzmann distribution: both A and B can go anywhere:
State 1 State 2 State 3
AB
AB
AB
A B
B A
A B
B A
A B
B A
• Bose-Einstein statistics: because A and B are indistinguishable, there is now only 1 possibility for the states where A and B would be in different states.
State 1 State 2 State 3
AA
AA
AA
A A
A A
A A
• Fermi-Dirac statistics: now states with two particles in the same state are not possible anymore:
State 1 State 2 State 3
A A
A A
A A
Both Bose-Einstein statistics and Fermi-Dirac statistics tend to the Maxwell-Boltzmann distribution in the limit of either:
TODO: show on forumulas. TODO experimental data showing this. Please.....

### Maxwell-Boltzmann distribution (MB distribution)

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#### Experimental verification of the Maxwell-Boltzmann distribution

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Most applications of the Maxwell-Boltzmann distribution confirm the theory, but don't give a very direct proof of its curve.
Here we will try to gather some that do.
##### Zartman Ko experiment
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###### Stern-Zartman experiment (1920)
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Is it the same as Zartman Ko experiment? TODO find the relevant papers.
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## Thermodynamics

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### Boltzmann constant (, )

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This is not a truly "fundamental" constant of nature like say the speed of light or the Planck constant.
Rather, it is just a definition of our Kelvin temperature scale, linking average microscopic energy to our macroscopic temperature scale.
The way to think about that link is, at 1 Kelvin, each particle has average energy: $$1/2kT (1)$$ per degree of freedom.
This is why the units of the Boltzmann constant are Joules per Kelvin.
For an ideal monatomic gas, say helium, there are 3 degrees of freedom. so each helium atom has average energy: $$3/2kB​T (2)$$
If we have 2 atoms at 1 K, they will have average energy , and so on.
Another conclusion is that this defines temperature as being proportional to the total energy. E.g. if we had 1 helium atom at 2 K then we would have about energy, 3 K and so on.
This energy is of course just an average: some particles have more, and others less, following the Maxwell-Boltzmann distribution.

### Thermodynamic potential

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chemistry.stackexchange.com/questions/7696/how-do-i-distinguish-between-internal-energy-and-enthalpy/7700#7700 has a good insight:
To summarize, internal energy and enthalpy are used to estimate the thermodynamic potential of the system. There are other such estimates, like the Gibbs free energy G. Which one you choose is determined by the conditions and how easy it is to determine pressure and volume changes.

#### Enthalpy ()

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Adds up chemical energy and kinetic energy.
Wikipedia mentions however that the kinetic energy is often negligible, even for gases.
The sum is of interest when thinking about reactions because chemical reactions can change the number of molecules involved, and therefore the pressure.
To predict if a reaction is spontaneous or not, negative enthalpy is not enough, we must also consider entropy via Gibbs free energy.

#### Gibbs free energy ()

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TODO understand more intuitively how that determines if a reaction happens or not.
$$ΔG=ΔH−TΔS (3)$$
At least from the formula we see that:
• the more exothermic, the more likely it is to occur
• if the entropy increases, the higher the temperature, the more likely it is to occur
• otherwise, the lower the temperature the more likely it is to occur
A prototypical example of reaction that is exothermic but does not happen at any temperature is combustion.
##### Reversible reaction
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I think these are the ones where , i.e. enthalpy and entropy push the reaction in different directions. And so we can use temperature to move the Chemical equilibrium back and forward.

### Entropy

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OK, can someone please just stop the philosophy and give numerical predictions of how entropy helps you predict the future?
The original notion of entropy, and the first one you should study, is the Clausius entropy.
For entropy in chemistry see: entropy of a chemical reaction.

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#### Second law of thermodynamics

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Subtle is the Lord by Abraham Pais (1982) chapter 4 "Entropy and Probability" mentions well how Boltzmann first thought that the second law was an actual base physical law of the universe while he was calculating numerical stuff for it, including as late as 1872.
But then he saw an argument by Johann Joseph Loschmidt that given the time reversibility of classical mechanics, and because they were thinking of atoms as classical balls as in the kinetic theory of gases, then there always exist a valid physical state where entropy decreases, by just reversing the direction of time and all particle speeds.
So from this he understood that the second law can only be probabilistic, and not a fundamental law of physics, which he published clearly in 1877.
##### Time reversibility
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###### Time reversibility of classical mechanics
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Considering e.g. Newton's laws of motion, you take a system that is a function of time , e.g. the position of many point particles, and then you reverse the speeds of all particles, then is a solution to that.
###### Time reversibility of gravity
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I guess you also have to change the sign of the gravitational constant?

### Phase transition

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TODO can anything interesting and deep be said about "why phase transition happens?" physics.stackexchange.com/questions/29128/what-causes-a-phase-transition on Physics Stack Exchange

#### Second-order phase transition

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The more familiar transitions we are familiar with like liquid water into solid water happen at constant temperature.
However, other types of phase transitions we are less familiar in our daily lives happen across a continuum of such "state variables", notably

### Refrigerator

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#### Dilution refrigerator

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Used for example in some times of quantum computers, notably superconducting quantum computers. As mentioned at: youtu.be/uPw9nkJAwDY?t=487, in that case we need to go so low to reduce thermal noise.
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### Temperature

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For scales from absolute 0 like Kelvin, is proportional to the total kinetic energy of the material.
The Boltzmann constant tells us how much energy that is, i.e. gives the slope.

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