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

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Why can $\beta$ not be linearly proportional to $T$, that is $\beta = constant \times T$?

$\beta$ in statistical mechanics is equal to $\frac{1}{k_BT}$ in in thermodynamics, but I do not understand why $\beta\propto T^{-1}$ instead of, say, $\beta\propto T$?
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5answers
6k views

What are distinguishable and indistinguishable particles in statistical mechanics?

What are distinguishable and indistinguishable particles in statistical mechanics? While learning different distributions in statistical mechanics I came across this doubt; Maxwell-Boltzmann ...
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1answer
1k views

For which systems is the equipartition theorem valid?

Under what conditions does a system with many degrees of freedom satisfy the equipartition theorem?
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3answers
573 views

Is there an upper limit to temperature in thermodynamics or statistical mechanics

In many presentations of statistical mechanics where we have a system of particles having mass, such as the molecules of an ideal gas, the temperature is often equated to the average relative velocity ...
3
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3answers
129 views

Statistical Mechanics - Distribution of Energies

Consider a state space $\mathbb{X}$. The probability density function under a canonical ensemble is given by the Boltzmann distribution $$\pi_{\mathbb{X}}(x)=\frac{e^{-\beta ...
3
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1answer
766 views

Fermi-Dirac distribution derivation?

I am trying to derive the Fermi-Dirac statistics using density matrix formalism. I know that $$<A>= Tr \rho A.$$ So I started from $$<n(\epsilon_i)>= Tr \rho n(\epsilon_i)=\frac {1}{Z} ...
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1answer
1k views

What is the difference between reversible and irreversible adiabatic expansion?

What is the difference between reversible and irreversible adiabatic expansion? Is it true that the work done by the gas is the same but the pressure applied externally differ between two process? ...
2
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1answer
160 views

Entropy is constant. How to express this equation in terms of pressure and density?

In hydrodynamics of an ideal, non-compressive flow we use 5 variables: pressure $p$, density $\rho$ and velocity field $\mathbf{v}$. So we need 5 equations. Landau's "Hydrodynamics" states that the ...
2
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2answers
154 views

Free electron gas in two dimensions

Can someone give a qualitative description on why the density of states for a two dimensional free electron gas is independent of energy while it is not in one and three dimensions? In one dimension ...
2
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2answers
788 views

RMS Free Path vs Mean Free Path

I am trying to determine the mathematical difference between mean free path and root-mean-square free path. For an ideal gas, the relaxation time is $$\tau=\frac{1}{\sqrt2 \pi nd^2 \bar v}$$ and the ...
2
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2answers
157 views

Bolzmann entropy [duplicate]

The Boltzmann entropy is defined as the logarithm of the phase space volume (E). Is there a reference, book, paper which shows where this definition comes and how it is equal to the phase space ...
2
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1answer
793 views

Probability of finding n particles in a volume v

I'm trying to calculate the probability of finding $n$ particles in a certain volume $v$ in a system with a total of $N$ particles and total volume of $V$. My problem is that I've tried two approaches ...
2
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3answers
784 views

Why the temperature is getting lower when the universe is expanding

As we know, if an ideal gas expands in vacuum, as its energy is unchanged, the temperature remains the same. An ideal gas's energy does not depend on volume. In general, the energy is $kT$ times the ...
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1answer
306 views

Scale invariance and self organized criticality

On wikipedia i have found this statement: In physics, self-organized criticality (SOC) is a property of (classes of) dynamical systems which have a critical point as an attractor. Their ...
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1answer
230 views

Equivalent system in Centre manifold theory

I was studying the centre manifold theory. It says (see Kuznetsov page 155, theorem 5.2) that the system on the left side of the picture is topologically equivalent to the one on the right. $ ...
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3answers
908 views

Definition of Fluctuations and Perturbations

The terms fluctuations and perturbations are frequently used in physics with different meanings. But they are confusing. Both terms seems to be same. Is there any one who can explain lucidly these ...
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2answers
134 views

Susceptibilities and response functions

It is often confusing whether a susceptibility is the same as a response function, specially that often they are used interchangeably, in the context of statistical mechanics and thermodynamics. Very ...
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1answer
76 views

Independent boson model with an arbitrary finite-dimensional impurity

The independent boson model consists of the following Hamiltonian: $$ H_s = E \sigma^z $$ $$ H_b = \sum_k \omega_k b^{\dagger}_kb_k $$ $$H_{sb} = \sigma^z \sum_k (g_k b_k + g_k^{\ast}b^{\dagger}_k).$$ ...
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3answers
113 views

When the low temperture reservoir with negative temperture (Kelvin), such as Ising model, is the efficiency of ideal heat engine larger than 1?

The ideal Carnot engine works between two heat reservoir with two temperatures $t_h$ and $t_l$. Its efficiency is then $1-\frac{t_l}{t_h}$ . If the low temperture reservoir is the Ising model with ...
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2answers
1k views

Must a reversible engine be a carnot engine?

I have this homework question: "Show that any reversible engine operating between T1 and T2 is a carnot engine." I think I have a solution, but it feels very hand-wavy. We know that any process that ...
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1answer
635 views

How is the distribution probability in the canonical ensemble derived?

I'm confused by the derivation of the canonical ensemble, namely the origin of the probability density, that is the Boltzmann factor. Here's what I have: We have a system of particles with ...
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1answer
136 views

Hamiltonian of a simple graph

I have a spin system: As shown in the picture, there are two spins S1 and S2, and a pair of interactions between them. One is a ferromagnetic interaction and the other is anti ferromagnetic ...
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0answers
45 views

About the factorial N! in the partition function

After reading these posts: Why is the partition function divided by $(h^{3N} N!)$? , What is the resolution to Gibb's paradox?, and some of these: http://arxiv.org/abs/1012.4111 , ...
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1answer
44 views

is it necessarily true that the partition function $Z$ (with degeneracies) $ =1$?

The partition function with degnerate energies is $$\text{Z}=\sum _ig_ie^{{-E_i}/{k_BT}}.$$ Because the partition function Z is defined as the normalisation constant, does Z always = 1?
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1answer
49 views

“Definition” of internal energy

Conversation of energy implies that if we have a thermally insulated system which goes from state 1 to state 2: $$\Delta E_{12}=E(2)-E(1)=\Delta W_{12}$$ and the 1st law of thermodynamics ...
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1answer
50 views

Definition of quantum microcanonical ensemble in Landau&Lifshitz

I'm reading the first chapters of Landau&Lifshitz 's [Statistical Physics][1] and I don't understand the definition of the quantum microcanonical ensemble. The microcanonical distribution for a ...
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2answers
82 views

Can one stimulate emission of a photon with an energy different from the emitted photon?

Suppose I have a three-level system with $E_0$ the ground level, $E_1$ the intermediate and $E_2$ the upper level. In thermal equilibrium they will have a certain probability distribution according to ...
0
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1answer
262 views

Math for Thermodynamics Basics

I am studying Statistical Mechanics and Thermodynamics from a book that i am not sure who has written it, because of its cover is not present. There is a section that i can not understand: ...
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2answers
581 views

Statistical interpretation of Entropy

I'm preparing my statistical physics course, and while writing the lecture notes it says that a system with non distinguishable particles has much less microstates asociated with a particular ...