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

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1answer
212 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: $$\frac{1}{N}\ln Z ~=~ \ln(2 \cosh^2(\beta J)) $$$$+ \frac{1}{2}\int_{-\pi}^{\pi}\frac{dq_1}{2\pi}...
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2answers
133 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
75 views

Estimate the persistence length of a rubber band [closed]

Not much more to say here, it's all in the question. The best, most convincing estimate will be chosen as the correct answer. EDIT: Assume the rubber band is at room temperature, with thickness $t$ ...
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0answers
59 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 ...
3
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0answers
224 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: ...
2
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1answer
80 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
144 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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0answers
243 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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0answers
55 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 ...
0
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0answers
70 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 ...
7
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3answers
432 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: $$...
0
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1answer
88 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 temperature)...
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0answers
71 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
115 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 ...
37
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6answers
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 ...
2
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1answer
237 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
64 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 ...
1
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1answer
218 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.
0
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1answer
225 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 ...
1
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1answer
163 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
54 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 ...
5
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0answers
86 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 $j$ ...
13
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2answers
464 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 ...
2
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0answers
200 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 ...
2
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2answers
2k 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 Maxwell-...
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1answer
54 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
2
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1answer
190 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
317 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
142 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). ...
3
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1answer
136 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} \, \mathrm{...
2
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1answer
162 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
254 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 ...
0
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1answer
221 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 H-...
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1answer
151 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. ...
2
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1answer
66 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 $\epsilon_n=(n+1/2)\...
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1answer
62 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.
0
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1answer
200 views

Summation to Integration in Statistical Mechanics

In Statistical Mechanics, what is the procedure of replacing this summation by the integration given by $$\sum_{\vec k} \rightarrow \frac{V}{(2\pi)^3} \int_{0}^{\infty} 4\pi k^2 dk$$
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1answer
545 views

Average Occupation Number in Bose Einstein Statistics using Grand Canonical Ensemble

If $Z=Z(z,V,T)$ is the Grand canonical Partition function, $\beta =\frac{1}{k_BT}$,$z=e^{\beta \mu }$ is the fugacity and $\epsilon_{\vec p}$ is the energy of a single particle in pth momentum state, ...
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2answers
2k views

What is the difference between classical thermodynamics and statistical mechanics? [duplicate]

What is the difference between classical thermodynamics and statistical mechanics? To me, they are greatly different but are different approaches for explaining same thing. But I do prefer ...
1
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1answer
62 views

Derivation of Fermi level for T>0

I am working through the derivation of the Fermi level $ \mu_0$ for T>0. However, at one point in the notes I have, it states without any explanation that: $$ \int_0^\infty F'(\epsilon) f_{FD}(\...
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1answer
61 views

Pressure in the grand canonical ensemble when momentum integration limits depends upon volume

When one does not want to consider the thermodynamic limit, it is possible in some systems to consider a dependance of the volume on the integration limits of the momentum. For example: $$\mathcal{Z} =...
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2answers
128 views

What is the solution for the apparent contradiction of Second law due to energy fluctuation?

A system has maximum entropy when it has reached thermal equilibrium. But as statistical mechanics say, there is always an otherwise infinitesimal probability of particles to confine at a corner of ...
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2answers
88 views

Magnons contribution to spontaneous magnetization

In Statistical Physics part II of Landau's course in theoretical physics it is stated that the magnon part of the spontaneous magnetization can be calculated as $$ M_m \equiv M(T) - M(0) = -\frac{1}{V}...
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1answer
2k views

Meaning of reversibility and quasistatic processes

A process in a closed system is reversible if the entropy change is $dS = \frac{dQ}{T}$. A process is quasistatic if a process is infinitely slowly. Now, if a process is reversible, this means ...
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1answer
2k views

Relation between isentropic/isenthalpic to adiabatic?

We have $dQ = T dS$. Does this imply that a process is adiabatic $dQ = 0$ if and only if it is isentropic $dS = 0$ for any process? This does not sound right, as this would mean that there is no ...
2
votes
1answer
305 views

Ideal chain / entropic spring - what is the *microscopic* force?

The ideal chain is the classic example of an entropic force. Usually one derives this force from the fundamental relation describing forces in the canonical ensemble: $$ \tag 1 F = (\partial \langle E ...
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2answers
214 views

Thermal Equilibrium and Canonical Ensemble

1 - Are two closed systems (with fixed volumes and of the same gas) in thermal equilibrium equivalent to two isolated systems at the same temperature? 2 - In the canonical ensemble, the "small ...
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0answers
69 views

Good Resource for Classifying Statistical Mechanics Problems [closed]

I've grown very interested in statistical mechanics ever since I took my first course in it. However, it feels like it is just overflowing with many types of problems and plenty of categories to ...
3
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1answer
367 views

Does Noether's theorem apply to entropy?

Entropy appears to have a translation symmetry - adding some constant value to it doesn't appear to my fairly rudimentary understanding of physics alter the actual physics. Is this correct? Now (...
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1answer
211 views

Definition of Entropy for reversible and irreversible process

$\int \dfrac{\delta Q}{T}$ can't be used to calculate entropy of an irreversible process. If you happen to know heat supplied and temperature at which it is supplied for just an instant. Can you then ...