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

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198 views

Flory-Huggins ternary phase diagram with a neutral component

I am searching the literature for the Flory-Huggins phase diagram with the following components : polymer, solvent, and a third component that does not interact with the other components (just entropy ...
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1answer
70 views

Entropy and chemical potential of an ideal gas

I am reading Schroeder's book "Thermal Physics". One calculation in the text was not quite clear to me. The entropy of an ideal gas is given by the Sackur-Tetrode equation: $$ ...
6
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2answers
76 views

Classical and Semi-classical treatments of the ideal gas

In the semi-classical treatment of the ideal gas, we write the partition function for the system as $$Z = \frac{Z(1)^N}{N!}$$ where $Z(1)$ is the single particle partition function and $N$ is the ...
4
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2answers
224 views

Grand canonical partition functions for Bose-Einstein statistics vs. Maxwell-Boltzmann statistics

In Bose-Einstein statistics, the grand canonical partition function is $$\mathcal{Z}=1+e^{-\beta(\epsilon-\mu)}+e^{-2\beta(\epsilon-\mu)}+e^{-3\beta(\epsilon-\mu)}+\cdots$$ In Maxwell-Boltzmann ...
3
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1answer
163 views

Why are there gapless excitations in the anti-ferromagnetic Heisenberg model while the true ground state is a singlet?

The true ground state of the anti ferromagnetic quantum Heisenberg Model (nearest neighbor only)is known to be a singlet (I think this is Liebs theorem.) Since a singlet is invariant under rotations, ...
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1answer
80 views

I'm getting weird autocorrelations when simulating an Ising model below the critical temperature

So I'm simulating an Ising model using Monte Carlo and the Metropolis algorithm. After letting it reach equilibrium, I try to calculate the autocorrelation of the magnetization. As long as the system ...
4
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0answers
72 views

Intuitively, why does removing solutes cost $k_B T$ of free energy per molecule?

I can calculate that if you want to, for example, desalinate water, you will have to pay a free energy cost of $k_B T$ for each ion you remove. In other words, removing an ion from a volume of water ...
5
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1answer
119 views

Energy fluctuations in quantum canonical ensemble

How would you go about showing that in the quantum canonical ensemble (that is, in the density matrix and operator formulation), the energy fluctuations, namely $\langle H^2\rangle - \langle ...
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0answers
49 views

Is entropy a dynamical quantity [closed]

I am confused whether Entropy is a dynamical quantity or not. Gibbs entropy, and quantum mechanical Von Neumann entropy.
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7answers
3k views

Homemade salad dressing separates into layers after it sits for a while. Why doesn't this violate the 2nd law of thermodynamics?

The oil, vinegar and other liquids in homemade salad dressing separate into layers after sitting for a while, making the mixture become more organized as time evolves. Why doesn't this violate the ...
0
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1answer
83 views

The “replica trick” initial formula?

In Spin-glass theory for pedestrians by Castellani and Cavagna, the initial formula used to introduce the replica trick is written as: $$\overline{\log ...
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2answers
237 views

Is there an equation to calculate the average speed of liquid molecules?

I seem to remember from first year physics that we can calculate the RMS speed of a stationary, ideal gas with $v=\sqrt{\frac{3RT}{M}}$. Does a similar equation exist for liquids?
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0answers
106 views

Understanding the product of partition functions by making sense of the maths and the physics

I have $N$ distinguishable particles in a 1D harmonic oscillator potential with 'proper' frequency $\omega$. The particles also have internal spin-$\frac12$ degrees of freedom in a magnetic field $B$ ...
8
votes
2answers
268 views

Chemical potential in Thermodynamics

In many scenarios, on computing the partial derivative of the internal energy (U) with respect to mole number (N) is negative. This implies that adding more moles of the substance decreases the U of ...
2
votes
2answers
178 views

Interpreting the Partition Function and Free Energy Mathematically

Given that The partition function in statistical mechanics tells us the number of quantum states of a system that are thermally accessible at a given temperature ...
3
votes
1answer
109 views

Definition of Information in Information Theory

I am not sure in which SE site I have to put this question. But since I have learnt Shannon Entropy in the context of Statistical Physics, I am putting this question here. In the case of Shannon ...
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2answers
221 views

Monte-Carlo and $O(n)$ models for non-integer n

$O(n)$ lattice statistical models can be generalized to non integer values of n, starting from their (expanded and resumed in graphs) partition function: $$Z = \sum_{\text{loop configurations}} n^{\# ...
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0answers
38 views

Two-dimensional atomic trap--how to set up the problem?

It is possible to trap neutral atoms between two solid surfaces in a potential of the form $V (x, y, z) = ax^2 + by^2$ where a and b are parameters. The allowed space for the gas extends to ...
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0answers
42 views

On the relationship between entropy and chaotic noise

I have few conceptual questions related to application of chaos in communications. Kolmogorov-Sinai Entropy1 , Kolmogorov-Sinai Entropy2 and Kolmogorov-Sinai Entropy3 KS is an entropy metric for ...
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0answers
26 views

Condensate fraction and single-particle density matrix

In Bose–Einstein condensation (BEC), how to prove the largest eigenvalue of the single-particle density matrix $$\rho_{ij}=\frac{\langle\Psi|a_i^{\dagger}a_j|\Psi\rangle}{N}$$ is ...
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2answers
65 views

Would it be possible to measure the change of entropy of a system? why?

To be more specific, what I mean is to measure it in a experiment. And if the answer is no, I want to know if it is principally impossible, or just impossible due to the technic limitation of our ...
0
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0answers
44 views

Landau free energy

I am reading the statistical mechanics by Pathria in Chap 12. I have a question about the Landau free energy. What is the physical reasoning for that the free energy could be a functional of the order ...
2
votes
1answer
90 views

What happens to the free energy of the two-dimensional ising model with vortices?

The classical 2d Ising model has a Hamiltonian of the form: \begin{equation} H = -\sum_{m,n = 0}^{M,N} J_1 x_{m,n}x_{m+1,n} + J_2 x_{m,n}x_{m,n+1} \end{equation} The partition function for the model ...
3
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1answer
118 views

Solving non-linear ODE for divalent solution at a 1-D surface boudary

I am trying to solve the following equation for a positively charged plane with charge density $\sigma$ at $z = 0$. $$ \phi''(z)=-\frac{e}{\epsilon \epsilon_0} \big(z_+n_{+} e^{-\beta z_+ ...
5
votes
1answer
58 views

Vanishing Planets?

If we put a solid sphere in space, it will lose some molecules which will form a sort of an atmosphere around it so that we have the required vapour pressure for solid-vapour equilibrium (Temp. of ...
5
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2answers
118 views

In thermodynamic systems why must the free energy of the system be minimized?

Is this somehow a consequence of the second law of thermodynamics?
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3answers
265 views

Is there an upper limit to temperature in thermodynamics or statistical mechanics

In many presentations of statistical mechanics where we have a system of particles having mass, such as the molecules of an ideal gas, the temperature is often equated to the average relative velocity ...
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0answers
69 views

Questions on degenerate ground states and the thermodynamic limit?

For example, let's consider a $N$ spin-1/2 system on a lattice described by the Hamiltonian $H$. My questions are: (1) If $H$ has either global $SU(2)$ spin-rotation symmetry or time-reversal ...
8
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1answer
122 views

What precisely does the 2nd law of thermo state, considering that entropy depends on how we define macrostate?

Boltzmann's definition of entropy is $\sigma = \log \Omega$, where $\Omega$ is the number of microstates consistent with a given macrostate. If I understand correctly, this means that it only makes ...
11
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2answers
402 views

Quantum entaglement and the arrow of time

I have seen several claims to that quantum mechanics is required to explain the arrow of time which I take to mean the macroscopic irreversibility of physical systems. This is presumably to resolve ...
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0answers
70 views

Ising model. What is large fluctuations of magnetization?

My background is in mathematics. I have studied the Ising model in $\mathbb{Z}^2$. The main model of statistical mechanics. Yesterday, I was reading the preliminary notes of the book Statistical ...
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0answers
59 views

Chemical potential of photons [duplicate]

Why do photons have zero chemical potential and what is its the physical significance? From what I know the chemical potential could be interpreted as the energy per unit particle that is put into a ...
2
votes
2answers
91 views

Free electron gas in two dimensions

Can someone give a qualitative description on why the density of states for a two dimensional free electron gas is independent of energy while it is not in one and three dimensions? In one dimension ...
0
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1answer
28 views

Probability distribution of two particle types system

Suppose that particles of two different species, A and B, can be chosen with probability $p_A$ and $p_B$, respectively. What would be the probability (and distribution) $p(N_A;N)$ that $N_A$ out of ...
2
votes
1answer
79 views

Statistical count

I am reading the book"Heat and Thermodynamics" by Mark Waldo Zemansky and Richard Dittman. In the chapter "Statistical Mechanics" it says if I have $N_{i}$ distinguishable particles in any of $g_{i}$ ...
3
votes
1answer
93 views

Is there a known equation for evolution of classical particle probability density?

Suppose we have some very imprecise knowledge of classical particle's coordinates and momentum: what we can only tell is the probability density to find it in some point of phase space. This is ...
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0answers
48 views

Is it possible to define a notion of temperature in a microcanonical ensemble?

I am thinking of a mircrocanonical ensemble as a finite system for which the number of particles, volume and the total energy is fixed. Is there a more refined view of this? Can I think of ...
3
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0answers
63 views

What real experimental systems are well-described by Glauber-Ising spins?

I'm hoping for references to actual physical systems in which all or at least most of the following can be simultaneously characterized: the spin flip rate, the temperature, and a relaxation or ...
2
votes
1answer
149 views

RMS Free Path vs Mean Free Path

I am trying to determine the mathematical difference between mean free path and root-mean-square free path. For an ideal gas, the relaxation time is $$\tau=\frac{1}{\sqrt2 \pi nd^2 \bar v}$$ and the ...
2
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0answers
87 views

Statistical mechanics of a coin toss

I'd like to ask some questions about flipping two coins related to statistical mechanics, e.g. microcanonical distribution, phase space distribution function etc... after I rephrase the coin flipping ...
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0answers
104 views

phase-space volumes or cells for N particle system

For N non interacting spinless particles in a volume, we have 3N degrees of freedom and we can divide the phase space into 6N dimensional cells of volume h raised to power 3N. And each cell ...
2
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1answer
112 views

Thermodynamics, chaperones : How to model polymer fragmentation

Living polymers are well described by equilibrium statistical physics. Now I would like to consider a case were living polymers undergo fragmentation due to chaperones. I can think of a kinetic ...
2
votes
1answer
108 views

Quantum Fourier Transform and Entropy

QFT is a nonlocal unitary transformation and so can generate entanglement in a system. It means a separable pure state can be converted into an entangled pure state. Now since the presence of ...
2
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0answers
61 views

Three-body correlation function in kinetic theory

In Kinetic Theory, one studies the evolution of a system of $N$ particles interacting with each other. We use the notation $\boldsymbol{w}_{i}$ to describe the coordinates in phase-space of each ...
3
votes
2answers
85 views

Statistical Mechanics - Distribution of Energies

Consider a state space $\mathbb{X}$. The probability density function under a canonical ensemble is given by the Boltzmann distribution $$\pi_{\mathbb{X}}(x)=\frac{e^{-\beta ...
3
votes
2answers
201 views

Connection between QFT and statistical physics of phase transitions

I have heard that there is a deep connection between QFT (emphasized by its path-integral formulation) and statistical physics of critical systems and phase transitions. I have only a basic course in ...
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2answers
46 views

Is there an analogue to the role of vapor in liquids and gases, but for solids and liquids?

It seems common for an ordered phase to have some amount of disorder present. For example, the average moment of a ferromagnet is less than maximum except at T=0 due to the presence of fluctuations. ...
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0answers
100 views

Boltzmann distribution: derivation from canonical distribution

I'm trying to understand the Maxwell-Boltzman Distribution, and in particular the derivation from the boltzman distribution for energy. I have successfully created an incorrect derivation, but I'm not ...
4
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1answer
81 views

A real gas with gravitation-like interaction

Consider a system (a gas) of point-like particles with a gravitation-like interaction (potential) $V(r) \sim \frac{1}{r}$ between pairs of them. One can rule out statistically that two particles will ...
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1answer
38 views

Can electrons coincidentally flow along a circuit to cause current?

My understanding of circuits which are not supplied an e.m.f. is that the electrons randomly just flow about in random directions, and since there's so many of them, probability dictates that any ...