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

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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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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. ...
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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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63 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.
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211 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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550 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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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 ...
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1answer
64 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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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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129 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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89 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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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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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 ...
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308 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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217 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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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 ...
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374 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
225 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 ...
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62 views

What is the central charge of the disordered $q$-state Potts model, for large $q$?

The central charge of a model, is, heuristically related to the number of microscopic degrees of freedom. Is there a simple argument for the asymptotic behavior of the central charge for the ...
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845 views

Efficiency larger than one?

The efficiency of a heat engine is the work we can do divided by the heat we take out of the hot reservoir. This quantity is always $ \le 1$. The efficiency of a heat pump is the heat we can release ...
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154 views

Is Boltzmann distribution contradicting with the fundamental assumption of statistical thermodynamics?

In equilibrium statistical physics the fundamental assumption of statistical thermodynamics states that the occupation of any microstate is equally probable (i.e. $p_i=1/\Omega, S=-k_B\sum p_i\,{\rm ...
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2answers
70 views

Why don't we have particles whose wavefunctions are symmetric wrt one exchange operator and anti-symmetric wrt other exchange operator?

Consider a system with $n$ identical particles. Let the wavefunction of the system be $\psi(r_1,\ldots, r_2)$. Let $P_{a,b}$ represent the exchange operator which exchanges particle $a$ with particle $...
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Physical meaning of RG transformation

When we do RG transformation in Statistical mechanics we eliminate unnecessary degrees of freedom and it leads us to the fixed point. How can I visualize it physically?
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71 views

How to derive the Bhatnagar-Gross-Krook collision integral from Boltzmann one?

Let's have Boltzmann collision integral: $$ I_{coll} =\int d \sigma d^{3}\mathbf p_{1}(ff_{1} - f{'}f{'}_{1})|\mathbf v_{rel}|.\tag{1}\label{1} $$ How to transform $\eqref{1}$ to BGK collision ...
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Free bosons with an attractive/repulsive defect

Consider a system of non-interacting bosons hopping in a qubic lattice in 2D or 3D. A single site of the lattice is an attractive/repulsive defect. Formally, let $H=-t\sum_{<i,j>}(a_i^\dagger ...
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Why are large scale structures isotropic in the Ising model?

I have at least a qualitative understanding of why the critical state of the Ising model is scale invariant, by arguments to do with renormalisation, which I understand only very roughly. However, in ...
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342 views

How is energy transferred in Joules law of heating?

Joule's law of heating states that an accelerated electron loses its energy, which is then converted into heat energy, by colliding with vibrating atom i.e ions in their lattice site. but we know atom ...
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73 views

Canonical Distribution (Partition Function)

For the canonical distribution $$ w_{n}=e^{(F-E_{n})/T}, $$ is the sum $$ Z=\sum_{n}e^{E_{n}/T} $$ a sum over energies or a sum over states? Perhaps this is a silly question, but Landau and Lifshitz ...
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363 views

Ising model 2-dimensional - ground state configuration

I have to prove something about the 2-dimensional ising model. The problem is the following: Prove that every nearest-neighbour and next-nearest-neighbour interaction on the square lattice $\mathbb{Z}...
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61 views

Simulating Phase Space Evolution

I am interested in modeling the time evolution of phase-space $\rho(\vec{q},\vec{p},t)$. I have attempted to use Liouville's theorem $\partial_t\rho=-\sum_{i=1}^{3}(\partial_{q_i}\rho)\dot q_i+(\...
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1answer
515 views

Why is molar specific heat at constant volume of a monatomic ideal gas a constant?

I thought specific heat varies depending on the substance. Why is it always $(3/2) R$?
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67 views

Applying Statistical Mechanics to Formulate Corrosion (Rusting)

I wanted to try and take my current knowledge of statistical mechanics (first quarter undergraduate course completed, beginning researcher in far from equilibrium statistical mechanics, basic ...
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155 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 ...
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42 views

How can one approximate integral def. of Z by the max value of the integrand?

I am taking a course in statistical physics, and while reviewing my notes from the lectures I came across something that I cannot get my head around. We arrive at an integral expression for the ...
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137 views

Subtraction In Quadrature?

I have a system of particles (electrons) with an initial RMS energy spread (say "1"). It goes through a section of constant magnetic field (bend magnet) and the electrons radiate. The electrons lose ...
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10k views

Change in entropy adiabatic expansion

I think that an adiabatic expansion of a gas should cause the entropy to increase. On the other hand we have for adiabatic processes that $dQ = 0$ and therefore $dS= 0$, which is why I thought that ...
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3answers
72 views

Change in energy ideal gas

I am supposed to calculate the change in energy upon changing both the temperature from $T_1$ to $T_2$ and the volume from $V_1$ to $V_2$. Now I was wondering whether this solution is correct: We ...
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2answers
81 views

From Quantum Mechanics to Statistical Mechanics in a Specific Case

I'd like to know how to get to statistical mechanics from the many-particle Schrodinger equation using a specific example, without using any Hamiltonian mechanics, phase spaces or ensembles, as a ...
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103 views

Does QM needs refinement?

Suppose atoms of an ideal gas are represented by non overlapping wave function so that the system can be described classically. As time passes the packets spread. Therefore over a period of time we ...
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164 views

First law of thermodynamics [closed]

In the first law of thermodynamics, we learned that $W$ and $Q$ are path-dependent quantities, but how are $Q$ and $W$ defined? I mean $W = \int_{\gamma} p(s) ds$ would be one possibility, where $\...
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199 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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386 views

Debye Model Density of States In One Dimension

I am trying to obtain the Density of states of the Debye model in one dimension I know the answer I am prepping for an exam and I am a bit stuck: The answer is: $\frac{L}{\pi*c_s}$ where $c_s$ is ...
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86 views

What is the density operator for an isothermal–isobaric ensemble (T,p,N)?

In the microcanonical ensemble $(E,V,N)$, the density operator is $$\hat{\rho}=\frac{\delta(\hat{H}-E\,\hat{I})}{Tr(\delta(\hat{H}-E\,\hat{I}))}$$ Where $\hat{H}$ is the Hamiltonian of the system and ...
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1answer
363 views

A simple experiment and the Maxwell-Boltzmann distribution

Consider two containers separated by a removable wall, each side of which is a perfect mirror for the gas in the respective container. Also the walls of the containers are ideal mirrors. In each ...
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1answer
91 views

chemical potential in BEC decreases in temperature

For a bose gas we can calculate the average number of particles through $$N = \int_0^\infty \rho(\varepsilon)n(\varepsilon) d\varepsilon$$ where $\rho(\varepsilon)$ is the particle density for energy $...
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Exorcism of Maxwell's Demon

I am possessed! Yes, with the thinking that if there is actually a Maxwell's Demon, then it would open the negligible weighted door which would ultimately make the second law invalid. But really can ...
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How does the movement of molecules change at the edge of a liquid?

I am thinking about how the velocity of molecules measured from a small region of space might change as the region of inquiry moves closer to the edge of a container. Ultimately I am thinking about MR ...
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122 views

What does Born Green equation signify physically?

What does Born Green equation obtained from YBG hierarchy for the equilibrium particle densities signify? I mean how can you model the equation into a physical problem?I understood the steps involved ...
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144 views

What it means to integrate over $n$ variables out of $N$, where $N>n$?

I was reading Theory of Simple Liquids, when I came across BBGKY hierarchy. In deriving the expression for the hierarchy, they integrate an integration of N variables over N-n variables to make the ...
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77 views

Maxwell's Inspiration to think about fields

I was looking at a Wikipedia article which had the following statement Atomists, notably James Clerk Maxwell and Ludwig Boltzmann, applied [...]. In modern literature Maxwell is often thought ...