# Tagged Questions

The study of large systems through coarse graining microscopic descriptions, providing a more detailed understanding of thermodynamics.

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### Could Navier-Stokes equation be derived directly from Boltzmann equation?

I know how to derive Navier-Stokes equations from Boltzmann equation in case where bulk and viscosity coefficients are set to zero. I need only multiply it on momentum and to integrate it over ...
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If I have a 2d random walk starting from origin, with equal probability of going in all four directions, what is the probability distribution of 'first' return to the origin?
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### Why is a random walk's RMS distance from the starting position$\propto\sqrt{N}$? [on hold]

Why is the root mean squared (RMS) distance from the starting position of a (1D, 2D, or 3D) random walk of $N$ equally-sized steps proportional to $\sqrt{N}$? Is it also$\propto\sqrt{N}$ for ...
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### Helmholtz energy in terms of grand partition function

According to my source of notes $$A = kT \left ( N \ln z-\ln Z(z, V, T)\right)$$ where $z=e^{\beta \mu}$, $\mu$ being chemical potential, $Z(z,V,T)$ is partition function for grand canonical ...
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### Casimir effect as an entropic force

When I first learned about the depletion interaction, my initial reaction was that it looks very similar to the Casimir effect. On making this remark to the professor, he replied somewhat mystically: ...
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### The Ideal Gas Equation in higher dimensions

Basically I wanted to know whether or not the ideal gas equation, $PV=NkT$ would hold in higher dimensions? If so, how would you go about proving this? I can't see any reason as to why it shouldn't ...
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### Example of a system where we cant measure the total possible microstates of a system but can measure the microstates of most probable state?

I have been reading about entropy and I read that entropy is basically a measure of total possible microstates but there was an approximation that when no. of particles become very large we take ...
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### Is there a phenomenon where physicists are only interested in the standard deviation of the quantity to be measured?

or a phenomenon where we can only measure the standard deviation ($\sigma_w$) of a variable $w$ and not the mean $\overline{w}$
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### Is there a way to get the Bethe Roots, that belong to a given eigenvalue of the transfer matrix?

(Quantum) integrable systems, that belong to solutions to the Yang-Baxter-equation, are often solved by the (algebraic) Bethe Ansatz. Solutions to the Bethe-equations lead to the eigenvalues of the ...
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### What does ''population'' mean in regards to the excited and ground states of an atom?

I have a problem I'm working on. It seems simple enough, but there is a term I'm not familiar with. It asks for the ''relative populations of the first excited and the ground states for helium gas in ...
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### Maxwell-Boltzmann distribution in Lennard Jones units

I'm studying thermostats in Molecular Dynamics. A very easy way (and poor, but I don't care for the moment) to implement a thermostat is to randomize momenta at some steps. These new momenta are ...
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### Nose-Hoover Barostat

Much can be found about the Nose-Hoover Thermostat. However I seem to be having difficulty finding out details about the Nose-Hoover Barostat, and how it is implemented. Would anyone be able to give ...
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### Physical interpretation of the chemical potential in Bose and Fermionic gas

I understand that both Fermions and bosons have the chemical potential $\nu <0$ when it is T>0, but still behave classically, the fermions would increase its chemical potential at T=0, whereas the ...
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### Microstates, Distribution of Particles, and the Probability of an Empty Compartment

If I have a closed system composed of $N$ particles and $p$ compartments, the total number of microstates available to that system is $$p^N$$ Now say I want to find the probability that any one of ...
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### Does fluctuation really occur in equilibrium as its microstates are allowed to occur by Fundamental Postulate in equilibrium?

The Fundamental Postulate says: In equilibrium, all accessible microstates are equally likely. Accessible means having same energy.(right?) Let a container is taken full of gas having number of ...
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### Classical Quantum or Relativistic?

An ensemble contains free electrons at 10^3 electrons per m^3 at 10^7 K. What can this ensemble be treated as: a Classical Quantum or Relativistic gas or in some overlapping domain?
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### How does the fundamental assumption of statistical physics make sense?

Consider two systems A and B in thermal contact. System A has $N_A=3$ simple harmonic oscillators and the system B has $N_B=3$ simple harmonic oscillators as well. Each system has a number of energy ...
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### Have we found a resolution to the Loschmidt paradox? [duplicate]

Loschmidt's Paradox (also known as the Reversibility Paradox) claims that it is not possible to deduce an irreversible process from time-symmetric dynamics such as the classic dynamics. This puts the ...
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### What is the cause for the inclusion of 'thermal equilibrium' in the statement of Ergodic hypothesis?

This is the fundamental assumption of statistical mechanics: In an isolated system in thermal equilibrium $^1$, all accessible microstates are equally probable. But why does it mention the ...
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### Why was the universe in a extraordinarily low-entropy state right after the big bang?

Let me start by saying that I have no scientific background whatsoever. I am very interested in science though and I'm currently enjoying Brian Greene's The Fabric of the Cosmos. I'm at chapter 7 and ...
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### How does temperature relate to the kinetic energy of molecules?

In ideal gas model, temperature is the measure of average kinetic energy of the gas molecules. If by some means the gas particles are accelerated to a very high speed in one direction, KE certainly ...
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### Reference for statistical mechanics from information theoretic view

I am interested in knowing if some one here knows book/notes for statistical mechanics from the information theoretic viewpoint.
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### Partion function for ideal gas - why use only one octant?

In these lecture notes (page 2) and in other sources I have checked, it says that the number of states with $k\in[k,k+dk]$ is: $$dN=\frac{4\pi k^2V}{8\pi^3}$$ Saying the factor of $8$ comes from the ...
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### Spontaneity / Free Energy of Non-Isothermal Process

I'm trying to determine a lower bound for the work input necessary to make an entropy-reducing process "spontaneous" in the sense that the 2nd law is not violated. For a constant temperature and ...
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### According to Liouville's theorem, why is the measure on an energy-surface different from the measure on the phase space in general

I recently read Khinchin's derivation of Liouville's theorem. I was able to follow the math for the most part, however I was hoping for an intuitive understanding about why the form of the measure on ...