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

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What are the Fermi and Debye temperature constants?

What are the Fermi temperature and Debye temperature constants? We were discussing these in class and I don't fully understand what these constants are or why we have them. Can anyone explain?
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1answer
68 views

Can you thermally pump a laser? (and problems with population inversion)

Recently a question was asked during a lecture about the possibility of thermally pumping a laser. The lecturer claimed that this is pretty impractical as typical transitions in the visible light ...
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173 views

Confusing Chemical potential of mixtures

I feel that there are very few textbook that treat the chemical potential of mixtures in an understandable clear way, which is why I wanted to ask here about certain things? Although I do not have a ...
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46 views

Is any phase associated with some fixed point in Renormalization Group?

In Wilson's paper I found a lot of discussion in expansions near a fixed point. He suggested that each fixed point is associated with a regime of the system. Like the fixed points of Anderson's Model, ...
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56 views

Ising Model with All Spins Interacting with All Other Spins

I am studying the Ising model with all spins interacting with all other spins and have formulated $Z$. I am trying to understand what it means to study at large N but not infinite N. I know that at ...
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1answer
106 views

How do sharp time intervals arise in a mesoscopic/macroscopic system?

$\newcommand{\ket}[1]{\left|#1 \right\rangle}$ $\newcommand{\bra}[1]{\left\langle #1 \right|}$ For a physical process in a mesoscopic/macroscopic system, how exactly can one deduce the time that ...
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24 views

Can a gas be modelled as a low density blackbody, if we want to consider how detectable it will be in space?

The answer to this question taught me about the sort of parameters I need to consider if I want to consider how "detectable" an object in space is. I want to consider the detectability of a ...
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2answers
70 views

Proof of the Boltzmann factor

The following derivation of the Boltzmann factor is obviously wrong, or incomplete: $$p(E) \propto \Omega(U-E)$$ Consider the Taylor expansion of $\ln\Omega(x)$: $$\ln\Omega(U-E)\approx \ln ...
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75 views

Resource recommendation: book that does not cover statistical mechanics specifically with thermodynamics in mind?

Statistical mechanics by its plain definition is a broad field, but most introductory textbooks focus on its applications in thermodynamics. Are there introductory texts that take up a broader view of ...
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32 views

Volume of highdimensional Sphere vs volume of spheres shell

When calculating the phase space volume $\Omega$ in the microcanoncial ensemble with fixed energy $E$, one integrates over a shell that includes all energies in between $E$ and $E+\delta E$: ...
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147 views

How hot is your photon?

This question comes from my answer to the question Can a cubic meter of space at absolute zero have any object with mass inside? and the related discussion under it. To summarize, I stated that the ...
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48 views

Entropy Inequalities

Hey I am reading this paper Entropy Inequalities by Araki and Lieb. I am trying to prove the following lemma: $$S^1+S^2\leq S^{12}+S^{23}$$ using the following lemmas: ...
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1answer
96 views

Deriving Boltzmann statistics from the maximum entropy principle

In some lecture notes I have, the author derives the expectation value of the occupation numbers for a discrete system of fermions as follows: Consider all states that have a certain energy ...
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16 views

Probabilistic interpretation of Hausdroff measure [closed]

My problem is to find/derive a pdf in terms of a parameter which closely resembles Hausdroff measure and the idea stems from the following concepts. Please correct me where I go wrong. Paper1 - ...
2
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1answer
136 views

Expansion of Onsager's Exact Partition Function for 2D Ising Model

We have a question where we are given the exact expression for the 2D Ising model partition function: $$\frac{1}{N}\ln Z ~=~ \ln(2 \cosh^2(\beta J)) $$$$+ ...
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1answer
65 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 ...
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0answers
50 views

How can energy be partitioned equally when energy is relative?

According to the Equipartition theorem in a system at equilibrium the energy should be on average be divided equally between the available degrees of freedom. The most common examples are the three ...
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142 views

Two-point correlation function for Potts Model

Consider the Potts model with three states , $\sigma (x) \in \{ 1, e^{2 \pi i/3}, e^{4 \pi i/3} \}$. I wanted to make sure that the following definition for two-point correlation function is correct: ...
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1answer
65 views

Different kinds of trace for statistical ensembles

In the chapter 7 of the book "A Modern Course in Statiscal Physics" by L. Reichl, we found $Tr[\hat{\rho}]=1$ for microcanonical ensembles and $Tr_N[\hat{\rho}]=1$ for canonical and grandcanonical ...
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1answer
110 views

Why is probability proportional to $ \ e^{-E/kT}$? [duplicate]

Why is the probability for say the Ising model to be found in state of energy E proportional to $e^{-E/kT}$ ? Is this some postulate or can it be derived from simpler principles?
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125 views

What are the fundamental “axioms” of statistical mechanics? [closed]

I have previously heard that some scientists are interested in trying to reformulate statistical mechanics in different ways to try and create new ways to solve novel problems. This got me wondering, ...
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33 views

Why is it inappropriate to calculate free energy change from end points alone?

In molecular dynamics, free energy changes are estimated using a variety of protocols to establish a path between the starting and ending states. The classic example is umbrella sampling in which a ...
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0answers
43 views

Brownian Ratchet Plausibility

Alright I'm going to throw whatever reputation I have on the line here. And yes this is a serious question. Apologies for the shoddy imagery. I had a couple ideas to get the Brownian Ratchet to ...
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3answers
231 views

How can I intuitively understand the Boltzmann factor?

It is known that for a system at thermal equilibrium described by the canonical ensemble, the probability of being in a state of energy $E$ at temperature $T$ is given by the Boltzmann distribution: ...
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1answer
75 views

Could the uncertainty principle theoretically be violated at 0 K? [duplicate]

Ok so please excuse me if the following mental argument is completely ridiculous or obviously flawed :P I was reading about how, even at 0 K (assuming we could experimentally reach such a ...
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38 views

Phase separation - density functional theory

I would like to get the equilibrium density profile $\rho(x)$ of a non ideal gas that has phase separated. I start by defining a simple free energy density. The total free energy $F[\rho]$ is a ...
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1answer
44 views

Molecular dynamics and detailed balance

In developing methods to perform Monte Carlo simulations one sufficient condition to preserve the stationarity of the target probability distribution is to impose detailed balance i.e. [Gardiner page ...
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5k views

Is there a thermodynamic limit on how efficiently you can solve a Rubik's cube?

Suppose I build a machine which will be given Rubik's cubes that have been scrambled to one of the $\sim 2^{65}$ possible positions of the cube, chosen uniformly at random. Is it possible for the ...
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1answer
164 views

Quantum ideal gas - Canonical ensemble - Occupation number summation notation (Huang)

(Question at the end, in bold, marked with an b)) For the quantum ideal gas, the hamiltonian (operator) of the system is: \begin{align} \mathcal H=\sum_{i=1}^N H_i=\sum_{i=1}^N \frac{P_i^2}{2m} ...
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0answers
46 views

Partition function for a two state system

We have a system of two energy states and we treat classical distinguishable and indistinguishable particles respectively. For the distinguishable case I thought that all in the left one one left ...
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1answer
56 views

Reversible and Quasi static processes

Do we have any proof that reversible processes are always quasi static or is it just a fact that hasn't been violated till date? If there is a proof then please provide a link.
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1answer
85 views

Sum over momentum states

In our lecture we used quite a couple of times that the sum over momentum states can be approximated by an integral over them. But instead of substituting $\sum_p \rightarrow \int d^3p$, we replaced ...
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0answers
23 views

Dimension of the Hilbert space of the restricted surface-on surface (RSOS) model

Right now I'm reading a paper on inversion identities for RSOS models, which you can find here. To give you a short introduction: The RSOS model is a face model, with a height variable assigned to ...
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1answer
75 views

Joule Thomson effect

I have difficulties to understand the Joule Thomson coefficient given on the wikipedia page. It says that $(\partial_p T) = \frac{V}{C_p}( T \alpha -1)$. Now my problem is that I don't know about ...
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1answer
40 views

Density depletion for Fermions

In my recent advanced statistical physics class, I read about the density depletion of Fermions, which are "defending" a given volume around them against other Fermions, while the exchange hole ...
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53 views

What is the argument for detailed balance in chemistry?

Detailed balance is an important property of many classes of physical systems. It can be written as $$ \frac{p_{i \to j}}{p_{j \to i}} = e^{\frac{\Delta G}{k_B T}},\tag{1} $$ where $i$ and $j$ ...
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254 views

Definition of stress at the microscale

Take, for simplicity, a Lennard-Jones fluid below the critical temperature, which is to say that there is a phase separation into fluid and gas and thus an interface is formed. The macroscale picture ...
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73 views

Why is the isothermal compressibility of the ideal boson gas larger than of the classical ideal gas?

Recently I came across (or well, derived in a lecture) the isothermal compressibility for an ideal boson gas. This was done in the context of statistical physics, using the quantum version of the ...
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2answers
513 views

Velocity Maxwell-Boltzmann distribution for dummies

I have a volume with N molecules; I need to assign to each particle a velocity vector: $$|\mathbf{v}_{i}|=[v_{x}, v_{y}, v_{z}]^{T}$$ for the i-th molecule; the velocities must fallow the ...
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1answer
33 views

Can a Fermi gas and a Bose gas be both at the same pressure and temperature?

The title says it all: can a Fermi gas and a Bose gas be both at the same pressure and temperature? It comes from a quiz about statistical mechanics
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1answer
110 views

Correlation length in d>1 Ising model, at zero temperature

I am studying the renormalization group approach to the Ising model using as a reference Cardy's book "Scaling and renormalization in statistical mechanics". I cannot understand what happens in the ...
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1answer
132 views

Meaning of chemical equilibrium between two phases

Suppose two phases 1 and 2 of water, say ice and water, are kept in a closed container, at a fixed temperature $T$ and fixed pressure $P$? Then I have the following question: Is phase 1 in ...
2
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1answer
77 views

Difference between collisional and collisionless Boltzmann equations?

Reading an excellent answer, I've read about there are different Boltzmann statistics for a collision-less system (f.e. stars in a galaxy) and in a system with collisions (f.e. gas in a closed box). ...
2
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1answer
51 views

Density of states and anisotropic distribution functions

We consider a $3D$ dynamical system. Its distribution function is given by the function ${ (\mathbf{x},\mathbf{v}) \mapsto f (\mathbf{x},\mathbf{v})}$, so that $$ \mathrm{d}^{3} \mathbf{x} \, ...
2
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1answer
101 views

Geometry, Group Theory, and Statistical Mechanics

During the course of my first statistical mechanics course we generally concerned ourselves with a bulk amount of our system and considered it in terms of a set of lattice sites that had a state. How ...
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1answer
152 views

Why is the canonical partition function an exponential?

It makes intuitive sense that micro-states of higher energy occur with a lower probability and the exponential function has reasonable properties. However can a physical explanation be given to why ...
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1answer
84 views

assumption of molecular chaos and the Loschmidt paradox

The assumption of molecular chaos says the velocities of two colliding particles are uncorrelated and also independent of time. Boltzmann actually used this assumption in his formulation of the ...
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1answer
104 views

What is the cause for mechanical equilibrium in statistical mechanics?

In classical thermodynamics, mechanical equilibrium is defined as the state of a system in which there is no net flow of volume as there should be no net pressure within the system. Ok. ...
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1answer
62 views

What's the closed-form of the sum relating to the DOS of simple harmonic motion?

In order to calculate the density of states of single particle in the simple harmonic potential, we would calculate that $$ D(\epsilon)=\sum_{n}\delta(\epsilon-\epsilon_n) $$ where ...
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1answer
46 views

Fermions and Bosons

For fermions $$P-\frac{Nk_BT}{V}\geq 0 $$ and for bosons, $$P-\frac{Nk_BT}{V}\leq 0$$ What can we understand from these results.