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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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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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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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27 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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42 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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22 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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2answers
48 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
81 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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2answers
68 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: Fundamental Assumption of Statistical Mechanics: For a system at ...
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1answer
27 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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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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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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2answers
51 views

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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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
35 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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57 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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2answers
99 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
55 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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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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2answers
58 views

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
140 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
73 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
72 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
50 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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0answers
41 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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0answers
19 views

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
58 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
22 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
20 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
52 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
54 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
65 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
135 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
36 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
54 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
157 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
40 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,...
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1answer
55 views

Maxwell relations for quantities per mole

Consider e.g. the differential of the Helmholtz free energy: $$dA = -SdT-pdV+\mu dn, $$ where, for simplicity, we only consider one particle species. We can then derive, for example, the following ...
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0answers
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What would a non-perturbative renormalization group treatment for polymers look like?

I know that one can do perturbative renormalization for the polymer excluded volume problem or the self-avoiding walk problem corresponding to n=0 component field theory. Here in Hamiltonian, we have, ...
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1answer
60 views

How to understand the two-point correlation function in momentum space?

Let's take the Ising model as an example and study the two point spin spin correlation function: $$\langle s_0 s_r\rangle = \frac{\sum_{\{s_i\}}e^{K\sum_{\langle i ,j\rangle}s_i s_j} s_0 s_r}{\sum_{\{...
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3answers
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Is the second law of thermodynamics a “no-go” theorem?

As defined here, there are several no-go theorems in theoretical physics. These theorems are statements of impossibility. The second law of thermodynamics may be stated in several ways, some of which ...
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1answer
74 views

Difference classical and statistical thermodynamics

What should I read to link between classical thermodynamics or engineerig thermodynamics and statistical mechanics?
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37 views

Total energy of a gas in an external magnetic field

Given the grand thermodynamical potential $\Omega$, the particle density $N$, the chemical potential $\mu$, the temperature $T$ and the entropy $S$, one can compute the total energy by the well known ...
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1answer
57 views

Screening of a coulomb-like potential

Imagine that we have a potential in the form of: $U(r) \propto \frac{1}{r^n}$ in a 2D system, with the high concentration of particles interacting with the above potential. How do you find the ...
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3answers
145 views

What mechanism at the microscopic level determines whether a system heats up or not?

When placed in an ordinary or a microwave oven, a beaker of water heats up except during boiling (i.e., a phase change involving latent heat). Now, suppose a system absorbs energy in such a way that ...
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1answer
45 views

Ising magnetization in metropolis

I am working on the Metropolis-Montercarlo algorithm for the square lattice ising 2D. Im running the simulations for a given lattice size, running from low temperature to high temperature, and ...