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

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12
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1answer
286 views

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 ...
-1
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0answers
50 views

2 D random walk first return to origin [on hold]

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?
1
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0answers
37 views

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 ...
0
votes
1answer
26 views

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 ...
10
votes
2answers
453 views

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: ...
1
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0answers
31 views

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 ...
1
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1answer
14 views

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 ...
4
votes
4answers
192 views

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 $\overline{w}$
0
votes
1answer
35 views

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 ...
1
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1answer
25 views

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 ...
0
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0answers
29 views

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 ...
3
votes
0answers
15 views

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 ...
3
votes
0answers
49 views

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 ...
2
votes
2answers
62 views

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 ...
0
votes
1answer
64 views

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 ...
1
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0answers
27 views

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?
1
vote
1answer
53 views

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 ...
7
votes
2answers
127 views

In a Monte Carlo NVT simulation How do I determine equilibration

I'm running an NVT (constant number of particles, volume and temperature) Monte Carlo simulation (Metropolis algorithm) of particles in two dimensions interacting via Lennard-Jonse potential ($U = ...
15
votes
1answer
2k views

Onsager's Regression Hypothesis, Explained and Demonstrated

Onsager's 1931 regression hypothesis asserts that “…the average regression of fluctuations will obey the same laws as the corresponding macroscopic irreversible process". (Here is the links to ...
0
votes
0answers
18 views

Understanding the concept of temperature vs. mean energy/heat capacity of a system

I need help understanding a concept in thermodynamics. What is the relationship between temperature and mean energy? What is the relationship between temperature and heat capacity? What I know: ...
2
votes
1answer
76 views

Gibbs entropy, Clausius' entropy and irreversibility

I have a bunch of doubts and confusions on the concept of entropy which have been bothering me for a while now. The most important ones are of a more technical nature, arisen from the reading of this ...
-3
votes
0answers
34 views

Free energy in magnetic materials [on hold]

The phenomenological description of magnetoelectrics and multiferroics is based on the series expansion of the free energy is powers of the magnetic and electric fields. WHY IS IT EXPRESSED IN TERMS ...
0
votes
1answer
21 views

Can collisions between particles in the canoncial ensemble be represented by a potential?

The professor in a statistical mechanics class said that in the canonical ensemble, you could represent interaction between particles as a potential in the partition function expression. But how about ...
3
votes
2answers
97 views

Why is entropy defined as a discrete sum over all microstates in classical case?

I'm reading about statistical definition of entropy, which says $$S=-k_B\sum_ip_i\ln p_i,\tag1$$ where $k_B$ is Boltzmann's constant, and $p_i$ is probability of $i$th state to be occupied. But in ...
5
votes
1answer
78 views

Why do we work in thermodynamic limit in statistical physics?

It is often stated that we work in thermodynamic limit at the beginning of courses on statistical physics $$N \to \infty, V \to \infty, \quad\frac{N}{V}=n=\textrm {constant}$$ what is less often ...
1
vote
1answer
122 views

Periodic ground state 1-dim ising model

Good evening! I'm at the beginning of my study about the Ising model and it has been proposed to me this problem: Find all periodic ground-state configuration for the following one-dimensional Ising ...
1
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0answers
13 views

Forward and backward work distributions in fluctuation theorem

Fluctuation theorems such as Jarzynski equality and Crooks theorem (Link), show that $\frac{P_f(W)}{P_b(-W)}=\,exp[\beta(W- \,\Delta F)]$ where $W$ is work done on the system during each ...
2
votes
5answers
1k views

Proof of Liouville's theorem: Relation between phase space volume and probability distribution function

I understand the proof of Liouville's theorem to the point where we conclude that Hamiltonian flow in phase-space is volume preserving as we flow in the phase space. Meaning the total derivative of ...
1
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2answers
122 views

Statistical mechanics vs. many-body theory

Where is the basic difference of statistical mechanics with many-body physics? What are the systems which cannot be studied in statistical mechanics but in many body theory? After all we know ...
1
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1answer
141 views

Entropy of the cosmological constant and the laws of thermodynamics?

Convention The convention being used is: $ A_{C} = $ The classical variable Premise Consider the following toy-model universe: A universe with a positive cosmological constant. Basic ...
4
votes
3answers
3k views

What is the meaning of Boltzmann definition of Entropy?

I would like to ask if someone knows the physical meaning of Boltzmann's definition of entropy. of course the formula is pretty straightforward $$S=K_b\ln(Ω)$$ but what in the heck is the natural ...
0
votes
1answer
50 views

CFT and temperature

I have tried to think about this for some time but could not really go anywhere. Sorry for the sloppy question and thanks for any pointer. My question is about CFT at finite temperature and ...
0
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0answers
20 views

Does ergodic hypothesis depend on the volume constraint of the macrostate or it only concerns with the energy constraint of the macrostate?

Ergodic hypothesis says that: [...] ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is ...
3
votes
2answers
196 views

Bose-Einstein condensation and phase transition

I would like to ask the following question for which I cannot find a definite answer in the literature. Of what ORDER is the phase transition leading to Bose-Einstein condensation for a ideal and ...
-1
votes
1answer
236 views

Dr. Pierre-Marie Robitaille: On the Validity of Kirchhoff's Law

Lately I've been researching about the black-body spectrum and the historical development of Planck's Law. I mainly wanted to understand a little bit more why many different objects (Stars, Hot ...
3
votes
1answer
182 views

Thermodynamics, chaperones : How to model polymer fragmentation

Living polymers are well described by equilibrium statistical physics. Now I would like to consider a case were living polymers undergo fragmentation due to chaperones. I can think of a kinetic ...
3
votes
1answer
103 views

Number theoretic loophole allows alternative definition of entropy?

A bit about the post I apologize for the title. I know it sounds crazy but I could not think of an alternative one which was relevant. I know this is "wild idea" but please read the entire post. ...
4
votes
1answer
674 views

Pauli paramagnetism for electrons with external magnetic field

Apparently it is to be shown that for electrons under an external magnetic field, in the limit as $B\to 0 $ $$ \chi = \frac{dM}{dB} \approx \frac{n\,\mu^{*^2}}{k\,T}\,\frac{f_{1/2}(z)}{f_{3/2}(z)} $$ ...
1
vote
2answers
100 views

Doesn't the success of statistical physics seem somewhat unreasonable?

It seems to me a rather big coincidence that statistical physics works so well. I can see how consistent macroscopic observations can occur just because the microstates that give rise to that ...
1
vote
1answer
132 views

Why is the logarithm of the number of all possible states of a system differentiable?

Temperature of a system is defined as $$\left( \frac{\partial \ln(\Omega)}{ \partial E} \right)_{N, X_i} = \frac{1}{kT}$$ Where $\Omega$ is the number of all accessible states (ways) for the system. $ ...
1
vote
0answers
31 views

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 ...
2
votes
4answers
160 views

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 ...
9
votes
6answers
6k views

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 ...
1
vote
3answers
3k views

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 ...
1
vote
1answer
35 views

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.
1
vote
1answer
31 views

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 ...
1
vote
1answer
149 views

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 ...
0
votes
2answers
110 views

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 ...
0
votes
0answers
14 views

Monte Carlo Metropolis method - trial step algorithm [migrated]

I'm working on a Magnetization simulation and writing an algorithm using the metropolis method. I am using a change in energy and a Boltzmann distribution, but, my question is about the trial step. ...
6
votes
0answers
164 views

Obtaining the canonical distribution from Fokker-Planck equation?

First I will provide a summary of the problem. Subsequently, I will provide more detail regarding the problem. Please note that entropy is in units of the Boltzmann constant. Summary I have a ...