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

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Chemical Potential in the canonical and grand canonical ensemble

I'm studying the ideal Fermi gas from "Statistical Mechanics", by R. K. Pathria. In particular, the following formula, which can be found on page 237: \begin{equation} \mu=\left(\frac{3N}{4 \pi g ...
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30 views

Computing Maximum Heat Capacity

I've encountered a problem in my statistical mechanics class that I'm not sure I'm approaching correctly: Consider a system of N interacting spins. At low temperatures, the interactions ensure that ...
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115 views

How can we define a velocity for quantum objects?

I have a question about quantum mechanics: I know that velocity is defined as the change of position with time, $v = \frac{\mathrm{d}x}{\mathrm{d}t}$. In quantum mechanics, the position of a particle ...
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51 views

Detailed balance of an energy function

I'm having difficulties understanding a problem about the acceptance rules for a given energy landscape. The Problem Suppose a system in which the energy is a function of x only: $$ e^{-\beta U(x)} ...
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1answer
42 views

Adiabatic transition from superfluid to Mott insulator?

I have a question about the dynamical passage from superfluid to Mott insulator state in the Bose-Hubbard model. Is it possible to go from superfluid region to the Mott insulator by changing the ...
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28 views

How to calculate the critical temperatures of noble gases such as He, Ne, and Ar?

It seems like Lennard-Jones potential should be used to calculate the critical temperatures of such noble gases. But I'm not sure how to actually do it. Could anyone give me some ideas or suggestions ...
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2k views

Why is entropy additive?

Although it seems simple, I can't get the derivation correct. Here is my reasoning: $P(S)=P(A)P(B)$ Where P is the probability and S, A, and B denote different systems. $S_A=-P(A)\ln P(A)$ and ...
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73 views

Derivation of Fermi-Dirac Distribution

How can I derive the Fermi-Dirac distribution function using simple mathematics? I am now tired of looking for the derivation on the net.So please help me to understand how actually electrons are ...
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53 views

Statistical Mechanics problem regarding the enthalpy and the expected value of energy

So I have an assignment(relating to a chapter on Canonical Ensemble) here with $H_E = \langle H\rangle$ where $H_E$ is the enthalpy, and $\langle H\rangle$ is the average of the Hamiltonian, I think. ...
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56 views

How to represent a Liouville projection superoperator in Hilbert space?

Is there a general way to represent a Liouville projection operator in Hilbert space, or can they take on any form so long as they satisfy the required properties of a projector? e.g. The thermal ...
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68 views

Why is the interaction energy of a dipole and a magnetic field *negative* when they are parallel?

The interaction energy between a magnetic moment, $\mu$, and an applied magnetic field, $B$, is given by $$\varepsilon=-\mu \cdot B$$ That negative sign is confusing my inuition. If we expand the ...
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24 views

Perodic boundary conditions vs Dirichlet?

I have been working through several examples recently involving particles in boxes (when finding the partition function of an ideal gas for example or looking at photon gases). I have seen two ...
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57 views

Why does the expansion of gas into a vacuum mean that we have less information about the system? (entropy)

I'm reading through Statistical Physics by F. Mandl and in the chapter about the 2nd law of thermodynamics he states that: The basic distinction between the initial and final states in such an ...
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84 views

How to justify the entropy maximum postulate using Statistical Mechanics?

The entropy maximum postulate states that given a thermodynamic system there's a function $S$ of the extensive parameters called entropy which has the property that once a constraint is removed the ...
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1answer
69 views

Density of states in a system of interacting electrons

When we are introduced to the density of states in typical band-theory problems we neglect interaction between electrons, and thus we define the density of states of a sigle particle as: ...
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62 views

Brillouin function - Classical Limit

The Brillouin function, defined as $$B_j(x) = \frac{j+1/2}{j} \coth\left(\frac{j+1/2}{j} x\right) - \frac{1}{2j}\coth\left(\frac{1}{2j} x\right),$$ tends to the Langevin funcion $$ \mathcal{L}(x) ...
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39 views

Entropy of fermi gas at $T=0$

Does the entropy of ideal fermi gas go to zero , in accordance with third law of thermodynamics? Consider a system of three fermions in a 3D box. The first fermion goes to the ground state of the ...
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57 views

Degeneracy of energy level, and density of states

Let there be a system in which the degeneracy of energy level does not depend on the energy and constant for each level, we denote it by G. And the density of energy states varies as $D(E)$. If ...
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11 views

How to interpret tunneling density of states in this simple model of the coulomb blockade?

I'm working on problem 3, chapter 6 in Altland and Simons' Condensed Matter Field Theory book. (Note: not homework.) We have two electron reservoirs and an "island" (the dot) which electrons can ...
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31 views

Confusion on Time and Ensemble Averages of Classical Harmonic Oscillator

Assume we have a classical harmonic oscillator $$ \ddot{x} = -k^2x.$$ Then the general solutions are of the form $x(t) = x_0cos(kt) + \frac{v_0}{k}sin(kt)$ where $x_0$ and $v_0$ are initial ...
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48 views

Applicability of Cardy's “doubling trick” to the 2D Ising Model

In Section 11.2.2 of the book on Conformal Field Theory by di Francesco, Mathieu, and Senechal (page 417), the two point function on the Upper Half Plane is written as being equal to the four point ...
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55 views

Statistical Physics of a System with Friction inside a Hot Bath

If you have a classical system (i.e obeying Newton's equations of motion) with Hamiltonian $H(x,p) = \frac{p^2}{2m} + U(x)$ then the statistical behaviour of this system is described by the ...
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108 views

Compute $\langle x^2\rangle$ for gas of non-interacting particles in a quadratic potential

Consider a 3d gas of $N$ non-interacting identical particles in a quadratic potential, where the Hamiltonian $H$ is: $$ H = \sum_{i=1}^N \bigg( \frac{p_i^2}{2m} + \frac{k}{2}x_i^2 \bigg). $$ After ...
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229 views

Why does Triple point exist?

In thermodynamics, the triple point of a substance is the temperature and pressure at which the three phases (gas, liquid, and solid) of that substance coexist in thermodynamic equilibrium. Is ...
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85 views

What does Liouville's Theorem actually mean?

Basically, the mathematical statement of Liouville's theorem is: $$\frac{\partial \rho }{\partial t}= -\sum_{i}\left(\frac{\partial \rho}{\partial q_i}\,\dot{q_i}+\frac{\partial\rho}{\partial ...
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45 views

Is the statement of “the fundamental assumption of thermal physics” given by Kittel incorrect?

I have just started re-reading Thermal Physics, by Kittel and Kroemer. They state the fundamental assumption of thermal physics as: ...a closed system is equally likely to be in any of the quantum ...
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20 views

Energy distribution in a gas coupled to a heat bath

I must be missing some point with regards to the canonical Distribution. Let us imagine I have a closed (to energy and matter) box full of ideal gas at temperature T. The total energy in the box ...
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37 views

How does $\rho(\dot{q_1}\mathrm dt)(\mathrm dq_2, \ldots,\mathrm dp_f )$ represent the no. of systems that would enter the volume in $\mathrm d t\;?$

I've been following Reif's Fundamentals of Statistical and Thermal Physics; there I came before the derivation of Liouville's theorem: There I couldn't understood few things. I could conceive the ...
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50 views

partition function of the U=0 Hubbard model

I'm trying to derive the following partition function for the U=0 Hubbard model: $Z=\prod_\mathbf{k}(1+e^{-\beta(\epsilon_\mathbf{k}-\mu)})$ My try was to use: $Z=\sum_{\sigma,\mathbf{k}} ...
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43 views

Is the formula of temperature $ \frac{1}{T}= \left(\frac{\partial S}{\partial U}\right)_{V,N}$ applicable to all type of ensembles?

I have seen multiple posts on this page that explained the statistical definition of Temperature as the derivative of the Entropy to the energy: \begin{equation} \frac{1}{T}\equiv ...
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108 views

Why does the chemical potential vanish for Bose Einstein condensate?

the reasoning in a Bose Einstein condensate is to try to account for all the particles in the excited continuum states by tuning the chemical potential. However at a critical temperature $T_c$ the ...
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49 views

Why is a random walk's RMS distance from the starting position$\propto\sqrt{N}$? [closed]

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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35 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 ...
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47 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 ...
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Example of a system where we can't 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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32 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 ...
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40 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 ...
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38 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 ...
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70 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 ...
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29 views

Classical Quantum or Relativistic? [closed]

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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69 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 ...
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109 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 ...
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31 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 ...
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22 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: ...
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84 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 ...
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27 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 ...
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93 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 ...
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175 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 ...
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66 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 ...
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1answer
130 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. ...