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Questions tagged [statistical-mechanics]

The study of large, complicated systems by means of statistics and probability theory, in order to extract average properties and to provide a connection between mechanics and thermodynamics.

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Thermodynamic Entropy seems to be contradictory

For an ideal gas the entropy change with energy is inversely proportional to temperature: This must yield: $$S=\frac 3 2 k_B \ln(T)$$ For various reasons, this equation is hard to find. However ...
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2answers
49 views

Question on the temperature dependence of the partition function

Let's just say we're looking at the classical continuous canonical ensemble of a harmonic oscillator, where: $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega^2 x^2$$ and the partition function (...
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10 views

$p$-spin spherical spin glass

Consider the $p$-spin spherical spin glass model with Hamiltonian $$H_{N,p}(\sigma)=\frac{1}{{N}^{\frac{(p-1)}{2}}} \sum \limits_{i_1,...i_p} J_{i_1,...i_p} \sigma_{i_1} \sigma_{i_2} .. \sigma_{i_p} $$...
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1answer
16 views

What is the sign of chemical potential of a noninteracting classical ideal gas obeying MB distribution?

The chemical potential of a noninteracting Bose gas can never be negative while that of a noninteracting Fermi gas can be both positive or negative. What can be said about the chemical potential of ...
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1answer
36 views

General formula for the variation of the chemical potential with temperature

For small temperatures $T$, such that $k_BT\ll \mu(T=0)\equiv \mu(0)$, the variation of chemical potential with temperature is given by $$\mu(T)=\mu(0)\Big[1-\frac{\pi^2}{12}\Big(\frac{k_BT}{\mu(0)}\...
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34 views

Math problem in Kaufmann-Onsager exact solution to 2D Ising model [on hold]

So, I've been following Huang book in Statistical mechanics for the 2D exact solution to the Ising model (chapter 15). During the solution he has to solve an eigenvalue problem, that is: $$(A+z_kB+...
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4answers
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How is quantum mechanics consistent with statistical mechanics?

Let's say we have an harmonic oscillator (at Temperature $T$) in a superposition of state 1 and 2: $$\Psi = \frac{\phi_1+\phi_2}{\sqrt{2}}$$ where each $\phi_i$ has energy $E_i \, .$ The probability ...
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What happens when we cool down the gas of non-identical particles?

For gas of identical particles, when we cool it down to extremely low temperature we can see one of two types of behaviour depending on the symmetry of wavefunction with respect to argument ...
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21 views

Conceptual meaning of Thermal States

Thermal states are generally defined as $$\tau(\beta)= \frac{e^{-\beta H}}{\mathrm{Tr}(e^{-\beta H})}$$ What are some physical statements one can make about them? A system in thermal equilibrium is ...
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2answers
30 views

How to see Planck's radiation law as a consequence of Bose Einstein statistics?

Planck's law comes about from the following ingredients. 1) The mode density per unit volume in a cavity is $8\pi\nu^2/c^3$. 2) Within each mode, assume Boltzmann statistics i.e the probability of ...
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What is the compressibility of this simple “book”? [migrated]

Compressibility is defined as $$C=\frac{2^{HN}}{2^{H_{max}N}}$$ The book is made up of a simple alphabet of only {a,b,c,d} which occur with probabilities $$P(a)=0.2, P(b)=0.4, P(c)=0.1, P(d)=0.3$$ ...
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Why do we take mean occupation number of particles? [on hold]

In statistical mechanics, when we find occupation number of particles using Bose-Einstein distribution or Fermi-Dirac distribution, why do we take the mean value of the occupation number? What does it ...
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21 views

The microcanonical ensemble approach to calculating the entropy of an ideal gas [duplicate]

I would like to set up the following problem. Assume I have a box of volume $V$ with $N$ noninteracting particles in it. The energy of each particle can be $\mathcal{E}_i$ such that $\sum_i \mathcal{E}...
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0answers
20 views

What is effective mean value? [on hold]

For counting the number of particles in any specific state we use effective mean value $\langle \hat{N} \rangle $ instead of number operator $\hat{N}$ in ensemble. I want to know what is the advantage ...
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2answers
33 views

How to write equation of state in terms of partition function?

While studying quantum gases (fermions, bosons ), equation of state written were $PV = k_B T Z_{gr}$, where $Z_{gr}$ is the partition function of grand canonical ensemble. P and V are pressure and ...
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1answer
44 views

Boltzmann distribution derivation from maximum entropy principle

I'm stuck halfway through a derivation of the Boltzmann distribution using the principle of maximum entropy. Let us consider a particle that may occupy any discrete energy level $\mathcal{E}_i$. The ...
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1answer
35 views

Increase of entropy as statistical necessity via Fundamental Assumption of Statistical Mechanics

My statistical physics books reasons that the increase of entropy for a closed system arises naturally from statistics. Outline: 1) Fundamental Assumption of Statistical Mechanics: For a system at ...
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1answer
26 views

Probability of a system in the canonical ensemble

In the canonical ensemble, we have the state of system $x_s$ and the state of the environment $x_e$. The probability of the total system is $$P(x_s,x_e)= const.$$ and that is independent of the states ...
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0answers
17 views

Detailed Balance Violation and Fokker-Planck Equation

Suppose I have a system with N sites, and each site can be modified (M) or anti-modified (A). Transitions between these two states are in part random, and in part auto-regulated by recruitment of At ...
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35 views

Must the whole universe have the same entropic arrow of time?

Theoretically, could it be possible for one galaxy to have low entropy in the past, but another nearby to have low entropy in the future? I understand that, cosmologically, there almost certainly are ...
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Whats the cross sectional area related to shear stress in torsion of a bar?

In shear force in a rectangular bar,the relevant area is the cross sectional area parallel to the applied force.But in torsion which also undergo shearing we get shear stress from torsion equation.I ...
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Deriving density of states in different dimensions in k space

The results for deriving the density of states in different dimensions is as follows: 3D: $g(k)dk = 1/(2\pi)^3 4 \pi k^2 dk$ 2D: $g(k)dk = 1/(2\pi)^2 2 \pi k dk$ 1D: $g(k)dk = 1/(2\pi) 2 dk$ I get ...
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44 views

Why are thermodynamic potentials minimized?

In thermodynamics one says that in equilibrium the corresponding thermodynamic potential is minimized. Why? For example take the case of a canonical ensemble. Based on the assumption that the ...
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1answer
32 views

Viscoelastic Constitutive Relation

In the Mori-Zwanzig formalism, the following identification for the generalised shear viscosity $\eta(t)$ is given: $$ \eta(t) = \frac{V}{k_B T} \langle \sigma(t) \sigma(0) \rangle, $$ identified as ...
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55 views

How energy $E= \pi k T$? [duplicate]

According to the Equipartition Theorem, $$E_{kin}= 3/2 k_B T$$ I read that when we are keeping the wave-nature of particles in mind, we can write $$E= \pi k_B T$$ But how we can write that?
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Order parameter and Bose-Einstein condensation

I want to study about order parameter and symmetry breaking related to bose einstein condensation in interacted system.which book i should read.also i want to learn this in second quantization ...
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31 views

Deviation from ideal gas [closed]

Can anybody help me solving this problem ? I am trying to solve this one by myself.
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2answers
82 views

Does quantum gases obey ideal gas equation $ PV= nRT$?

At extremely low temperature, does an ideal gas of bosons or fermions obey the ideal gas equation, $PV= nRT$?
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1answer
52 views

Can Einstein relation be used to calculate mobility under equilibrium dynamics?

In weak field non-equilibrium dynamics, mobility can be calculated by Einstein relation $\mu=\frac{eD}{K_BT}$, where $D$ is diffusion constant. Mobility can also be calculated by the definition $\mu=...
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14 views

Aggregation phenomena : How to get from a discrete to a continuous point of view

I'm studying a diffusion limited aggregation phenomenon. The $N$ particles diffuse in a box and when there is a contact they stick with a probability $p$, and let's say to simplify $p=1$. Meaning that ...
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What is the equation of state of foam, on a macroscopic scale?

Consider a large amount of soap foam (or any other substance producing foam), of mass density $\rho$ in a gravityless environment. What is the internal pressure $p$ of that foam, viewed as a fluid on ...
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Kinetic Theory of Solids

Is there a clean way to examine temperature for solids and liquids in classical mechanics like the kinetic theory for gases? I'd like to get a good explanation that doesn't involve much in the way of ...
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1answer
124 views

Interpretation of the Boltzmann factor and partition function

$$p_i = \frac{ \exp\left(-\frac{\epsilon _i}{k_BT} \right)}{Z} $$ $$ Z= \sum_{i} \exp\left(-\frac{\epsilon _i}{k_BT} \right)$$ A) Is $p_i$ the probability of the system having an energy equal to $\...
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Wolff cluster update in Monte Carlo simulation - at critical temperature [closed]

A general question to the Monte Carlo experts. When I use Wolff algorithm for global updates, say for Ising 2d, I always flip at least one spin (the initial random spin in the cluster). So, near the ...
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2answers
56 views

Mermin Wagner theorem proof, what does the K stand for ?

I've been reading about the Mermin-Wagner theorem recently. I think I understand pretty much every computation need to derive its result from the Bogoliub inequality, but there is one thing I don't ...
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1answer
67 views

How to explain imaginary kinematic viscosity of a vacuum?

According to the connection between the Schrödinger equation and the Navier-Stokes vacuum has the imaginary kinematic viscosity $\frac{ih}{2m}$. How to explain it? For the formation of the viscosity ...
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2answers
49 views

Alternate definitions of Thermal states

The definition of thermal states I'm used to is: $$\tau_{\beta} = \frac{1}{Z}\,e^{-\beta H}$$ where $Z$ is the partition function defined as $Z= \mathrm{Tr}(e^{-\beta H})$, $\beta$ the inverse ...
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34 views

Finding ground state energy using numerical real space renormalization group

I want to find ground state energy (as well as wavefunction) for spinless $tV$ model using Real-Space Renormalization Group (RSRG) approximation. The $tV$ model is defined as $$H=H_t+H_{int}=-t\sum_{i=...
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Sherrington-Kirkpatrick model with negative mean $J_0$

In the Sherrington-Kirkpatrick (SK) model, one considers an Ising Hamiltonian $$H = -\sum_{i<j}J_{ij}s_is_j$$ where $J_{ij}$ are drawn independently from a Gaussian distribution with mean $J_0$ ...
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1answer
48 views

Multiple Triple Points

I was reading Chandler's Introduction to Modern Statistical Mechanics and noticed a strange feature in one of the figures. The phase diagram in the image has two triple points; however, according to ...
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1answer
20 views

What Are the Correct Energy Levels for Computing a Molecular Partition Function with HITRAN?

I was trying to compute some stat mech and thermodynamic quantities using the data in the HITRAN molecular data base and ran into a conceptual problem. The basic quantity needed for these ...
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1answer
13 views

Thermodynamic functions of state for freely jointed polymer chain derived from partition function

I'm reading a stat thermo text (Terrel Hill) about the freely jointed chain problem. It all goes well until I hit the thermodynamic function of state derived from the canonical partition function. The ...
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2answers
45 views

Why quasistatic doesn't imply reversible process? [duplicate]

Quasistatic process are almost always in equilibrium. We know that equilibrium implies zero entropy change. And zero entropy change implies that the process is reversible. So why quasistatic doesn't ...
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1answer
51 views

Population of conformers in $NVT$ ensemble

I have one question - I've done simulations using Car Parrinello Molecular Dynamics together with Thermodynamic Integration method (liquid phase), hence I have calculated Helmholtz energies for my ...
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1answer
61 views

Estimating the inter-molecular van der Waals' force from the boiling point of water

Background: I understand that inter-molecular van der Waals' forces are responsible for maintaining water in the liquid phase. Now, if we suppose that the net van der Waals' force on a given H2O ...
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1answer
94 views

Density of states for 3D simple harmonic oscillator

I have the thermal partition function and the density of states for the 3D simple harmonic oscillator, which are given below $$ Z(\beta) = \frac { 1 } { \left( 2 \sinh \left( \frac { \beta \omega } { ...
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1answer
27 views

Can one formulate a fluctuation-dissipation theorem in presence of non-Gaussian noise sources?

The fluctuation dissipation theorem relates the linear response of a system to Gaussian fluctuations. The natural question that comes to my mind is the possible derivation of an analogous FDT in ...
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1answer
42 views

Fitting an Ising Model with Probabilities

Question How to use the observations to fit an Ising model? After self-studying for several days, my current guess is: $\theta_{ii} = \log[P(X_{i} = 1)]$ $\theta_{ij} = \log[P(X_{i} = 1, X_{j}=1)]$ ...
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2answers
143 views

The mean of Langevin equation

I have a very basic question regarding the mean of the Langevin equation. So we have an equation of the form: $$\dot{v}(t)=-\beta v(t)+ \xi (t)$$ Where $\xi (t)$ is a Gaussian white noise with an ...
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1answer
29 views

Convexity/Concavity of Thermodynamic Potentials and Entropy Maximization

In a brief review of thermodynamics, our lecture notes read Thermodynamic potentials are concave in their extensive variables and convex in their intensive variables. Alright, we start with $U(S,...