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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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Why does the classical continuous partition function blow up as $T \to 0$?

At $T = 0$, we'd expect Entropy to be zero because there's only one microstate and the $\log(1) = 0$. However, when I take the limit as $T \to 0$ in the classical canonical ensemble, it goes to ...
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34 views

What is the physical meaning of time-averaged Hamiltonian?

I saw in the literature related to the systems with periodic driving forces, people often define a "time averaged Hamiltonian" as $$H_{\text{avg}}=\frac{1}{T}\int_0^T H(t) \ dt.$$ But I do not ...
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30 views

Van der Waals fluid: density fluctuations [on hold]

I am given the following problem: Consider a region of spatial extent L (and with volume $L^3$) within a Van der Waals fluid given by the equation of state \begin{align} \beta p = \rho/(1-b\rho) ...
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1answer
30 views

Free energy along a reaction coordinate

I've come into this issue when trying to understand biased sampling methods, in particular, umbrella sampling, but I think the question is more general. A recurring argument is that, along a reaction ...
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23 views

RG of 2D Ising with nonzero magnetic field on triangular lattice

I am given the Ising Hamiltonian \begin{align} H = K \sum_{<ij>}S_i S_j + h \sum_i S_i, \quad K>0 \end{align} to set up a real-space block-spin RG, where the renormalized spins are ...
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21 views

Derivation of density of states for a gas with $N$ states

I am trying to find any information on the derivation of the density of states for a system with periodic boundary conditions in 3D. I know how it works with 1 particle since I have seen the ...
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2answers
70 views

Does $PV\propto T$ apply to a photon gas?

For an ultrarelativistic ideal gas, I know that $p=\frac{u}3$; $TV^3 =$ constant; $pV\propto T$. For a photon gas, I know that the first two results apply as well. However, I am unsure if the third ...
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43 views

Eigenvalues of the Hamiltonian

Is every eigenvalue of the Hamiltonian a form of energy? If not are there values of the Hamiltonian that do not correspond to the energy of the system?
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36 views

Sound Waves and the Boltzmann Distribution

Imagine a sound wave traveling linearly in a given direction through a monatomic ideal gas. Based on gas laws and the wave equation, we have that the wave should travel, in that given direction, at a ...
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1answer
26 views

Bond order correlation function

I am trying to compute the bond order correlation function, $g_6$. It is defined based on the bond order parameter: $$\psi_6(x_i) = \frac{1}{N_i}\sum_{i=1}^{N_i}{\exp(i6\theta_i^j)}$$ where $\theta_i^...
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Lagrange multipliers in Maxwell-Boltzmann statistics

I'm following Wikipedia's derivation of Maxwell-Boltzmann statistics. After applying Lagrange multipliers, we arrive at this expression for energy: $${\displaystyle E={\frac {\ln W}{\beta }}-{\frac {...
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1answer
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Statistical Mechanics & Dynamical Systems

As a (theoretical) physics student I've taken (advanced) undergrad courses in both statistical mechanics and dynamical systems (which was purely mathematical, treatment of nonlinear differential ...
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27 views

How is heat dissipation rate the product of force and velocity?

Let $q$ be heat dissipation to midium, $F$ be the force to a particle, and $\dot{x}$ is the velocity of it. According to the equation (8) in Seifert 2005, $\dot{q} = F \dot{x}$ holds. How does this ...
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2answers
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E. T. Jaynes' subjectivism vs measurement of distributions

In his paper, E. T. Jaynes argues that entropy is a measure of our ignorance about a system. As such, the probability distribution of states $\{p_k\}$ has to be chosen in the most unbiased way, thus ...
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2answers
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What is the density profile in the context of statistical mechanics?

While doing exercises in Statistical Mechanics I came across the following definition of the Density Profile of a system of $N$ non interacting particles $$\rho(\mathbf{r}) = N\langle \delta(\mathbf{...
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Please help me in solving this question [closed]

An atom has two energy levels with a transition wavelength at 5800 Å. At room temperature 4x1020 atoms are in the lower state. How many occupy the upper state under the conditions of thermal ...
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1answer
29 views

Can you integrate internal energy to get original partition function?

If I have the hamiltonian of the simple harmonic oscillator $$H = \frac{p^2}{2m} + \frac{1}{2} m \omega ^2 x^2 $$ Then it's partition function is: $$Z = \frac{k_b T}{\hbar \omega} $$ You can get ...
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Finding parameters by principle of maximal entropies of two independent systems sharing parameters

I have $n$ observations of system $A$, and $m$ observations of system $B$. $n \ne m$ Behavior of system $A$ and $B$ are independent, given the knowledge of the parameters. However, the two systems ...
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22 views

Why is it that the multiplicity function of two subsystems reaches a maximum if the $$\frac{\partial E_{1,2}}{\partial T_{1,2}}_{N,V} > 0$$

Intuitively it's clear to me and I understand why but when I try to break it down rigorously in a way that deals with the equations and their dependencies and derivatives I'm not sure that I can. ...
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1answer
26 views

Integral with a correlation [closed]

I was deriving the expectation value of position based probability distribution, and now need to derive $$\int x^\prime\omega_\text{eq}(x^\prime)[x^\prime-\langle x\rangle_\text{eq}]\mathrm{d}x^\...
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How to quantify frustration for spin models with long range interactions?

Consider the following Hamiltonian: $$ H=-\sum_{i\neq j}J_{ij}S_iS_j-\sum_i H_iS_i $$ where $S_i\in\{-1,1\}$, and the summed pair $i,j$ can be any two distinct indices (not necessary adjacent spins)....
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1answer
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Problem with finding the density of states of an $N$-body system

I am having problems solving a particular problem in my Statistical Mechanics course. We have a system that is composed of $N$ non-interacting particles each of mass $m$. The particles are bound to ...
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1answer
40 views

Hamiltonian of a quantum heat bath

I have seen the Hamiltonian for a heat bath written as: $$ H_B = \hbar \int_0^\infty \omega b(\omega)^\dagger b(\omega) d\omega $$ I was hoping to understand this equation better. This suggests that ...
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1answer
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What is the meaning of $\lim_{x\to0}\frac{\ln|-x|}{\ln|x|}$? [closed]

I am now working out some critical exponent, and I encountered this result $$\lim_{x\to0}\frac{\ln|-x|}{\ln|x|}.$$ Can I write this equals to 1? Here $x=\frac{T-T_{c}}{T_{c}}$ and $T_{c}$ is the ...
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Why particles should be indistinguishable in statistical mechanics when deriving Maxwell distribution? [duplicate]

When calculating the number of microstates in statistical mechanics, we assume that particles should be indistinguishable. What is the reason behind it?
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2answers
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Change of entropy in irreversible process

When calculating entropy change for a irreversible process,do I assume a reversible path and then integrated it?
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23 views

How large does a system have to be for $\Omega$ to be the multiplicity of the most probable macrostate?

Generally speaking, how large does a system have to be for the total multiplicity of the system to be (100.000000000000000000000000001 percent of) the multiplicity of the most probable macrostate? I ...
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1answer
22 views

Why is the internal energy the expected value of energies of individual particles?

In this Wikipedia page: https://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics) .. the total sum of energy in an ideal gas is given as: $$\langle E \rangle = \sum_s E_s P_s $$ But ...
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1answer
52 views

Entropy production, local thermodynamic equilibrium and adiabatic process

It is said that for local thermodynamic equilibrium the local entropy production needs to be 0. Now, I am reading the following from the book by de Groot and Mazur "Non-Equilibrium Thermodynamics". ...
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61 views

Partition function in the non-interacting limit

Let's consider the partition function $$Z(\lambda)=Tr (e^{-\beta H})=Tr (e^{-\beta (H_1+\lambda H_2)})$$ for a quantum system with the Hamiltonian $H=H_1+\lambda H_2$ where $H_1$ is the free part of ...
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1answer
32 views

Minimisation of Gibbs/Helmholtz free energy and Clausius theorem

I am trying to understand why (under the relevant given conditions) the free energy (either Gibbs or Helmholtz) is minimised. The derivation I have seen in several places goes like this. Set $\delta W ...
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Non-uniqueness of the Order Parameter and its Critical Exponent

In the theory of phase transitions, an order parameter is usually defined as some quantity which distinguishes the two phases of the system by being zero in one phase, and non-zero in the other (see e....
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Ideal gas as balls in boxes in different ensembles

I have some questions about the ideal gas in different ensembles. Often boxes with moving balls are used to explain an ensemble. Now I am confused what the difference would be in the corresponding ...
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Partition function of an asteroid gas (gravity)

Consider the classical problem (Newtonian gravity) of a large number of $N$ identical non-interacting asteroids orbiting around a big planet. I wanted to see if the problem was solvable. I wrote my ...
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Modelling liquids like water with BBGKY-hierarchy

The BBGKY hierarchy is a well-known useful possibility to derive kinetic equations for gases and Plasma. The N-particle System is reduced to few-particle Systems by Integration over many Phase space ...
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1answer
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Taking moments of the Vlasov equation

Given the following term: $\nabla_{v} \cdot \left[ \frac{e}{m_{s}} (\textbf{E} + \textbf{v} \times \textbf{B})f_{s} \right]$ where $\textbf{E}$ and $\textbf{B}$ are the electric and magnetic fields ...
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How do you experimentally measure the chemical potential of a gas?

How does one measure the chemical potential of a substance/ thermodynamic system? I am asking this question for two reasons: (1) The measure on phase space: Textbooks typically state that one should ...
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1answer
75 views

Partition function in spherical coordinates

Suppose I write the Hamiltonian/energy of my system in spherical coordinates ($r,\theta,\varphi$) with conjugated momentums($p_r,p_\theta,p_\varphi$). How do I calculate the partition function? If ...
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1answer
35 views

Wiedemann-Franz law derivation book recommendation

Can you recommend a good book with a thorough derivation. I know I'm more likely to find in a condensed matter book or a book on conductors but any recommendation would be appreciated, bonus if it ...
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0answers
25 views

Correlation length amplitudes in Ising 2D model

I am reading the article about Universal amplitude ratios in the 2D Ising model (https://arxiv.org/abs/hep-th/9710019) by G. Delfino. I have a question about page 3 of the paper. For a magnetic ...
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1answer
59 views

Continuous Transition of Degrees of Freedom in Thermodynamics With Simple Example

In thermodynamics books I have read, I have often come across statements about how certain degrees of freedom are relevant only at certain temperatures (such as the vibration degrees of freedom of ...
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1answer
131 views

Interpreting heat using information entropy

In an answer to the post about Microscopic Definition of Heat and Work, Ronan says, $$<dE> = \sum \epsilon_idp_i + p_id\epsilon_i$$ We can see that the change in average energy is partly ...
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1answer
75 views

What is the relation between chemical potential and the number of particles?

Chemical potential is defined as the change in energy due to change in the number of particles in a system. Let we have a system which is defined by the following Hamiltonian: $$H = -t \sum_i^L c_i^\...
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1answer
28 views

Integral limits in phase space

If I am calculating the partition function for $H=cp$, ultrarelativistic gas in three dimensions. And by breaking down $d \Gamma$ into $dq$ and $dq$ and further using spherical coordinates I will get $...
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0answers
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How bad is it if we don't know the distribution of an average?

Let's assume that it takes on average $\langle W\rangle$ work to perform some process. While we do know that the fluctuations i.e. the difference between single realizations of the process $W_1$,$...
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1answer
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Liouville equation with Dirac delta as probability density

I would lke to know if the probability distribution given by $$\rho(q,p,t)=\delta(q-q(t),p-p(t)) $$ with the initial condition $\rho(t=0)=\delta(q,p), $ where $q(t)$ and $p(t)$ are trajectories ...
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1answer
45 views

What is the order parameter of 2D generalized $XY$ model?

I'm now studying the phase transition of 2D generalized XY model. This model considered here has a mixture with ferromagnetic and nematic-like interactions, $$\mathcal{H}=-\sum_{\langle i j\rangle}\...
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1answer
45 views

Why can we say that at zero absolute temperature is there only one accesible state?

While studying the microcanonical ensemble, the entropy definition requires that at T=0 there is only one accesible state so that the entropy, S=0. Why is it true?
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1answer
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Phase separation in physics

I would look to familiarize myself with the current literature of phase separation. If one can direct me to statistical/thermodynamics theories of phase separation. Has phase separation been ...
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1answer
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What is role of nucleation centre in the formation of ice?

Water must be impure so that the impurities can act as nucleation centre for ice to form. What is role of nucleation centre? Why can not ice form with some nucleation centre.