# Tagged Questions

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

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### Helmholtz Free Energy, Partition Function

I'm trying to develop some basic intuition here, so this comes mostly as a jumble of commentary/questions. Hope its acceptable. Helmholtz Free Energy: $A = -{\beta ^{-1}}lnZ$. I find this statement ...
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### Parameter determining argon phase

Currently I am working a molecular simulation to determine phases of an argon NPT ensemble using Lennard Jones potential. Mainly I use the radial distribution function to determine solid, liquid, or ...
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### Computing microstate probabilities based on Boltzmann distribution for chemical systems - Is it rigorous?

One approach to predicting the folded structure of a polymer (DNA, RNA, protein) is to compute the probability that any particular part of the polymer $x_i$ is "paired" with another part of the ...
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### helmholtz free energy of a polymer

You have a polymer chain of $N$ units, which is represented by $N$ independent springs in series. The springs are Hookean, with spring constant $L$, and the end to end vector is $\mathbf r$. So the ...
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### Phase volume contraction in dissipative systems

I am confused about phase-volume contraction in dissipative systems. Please help me catch the flaw in my understanding. From a macroscopic point of view I understand that a dynamic system tends to go ...
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### Fermi's Golden Rule and Density of States

I know Fermi's Golden Rule in the form $$\Gamma_{fi} ~=~ \sum_{f}\frac{2\pi}{\hbar}\delta (E_f - E_i)|M_{fi}|^2$$ where $\Gamma_{fi}$ is the probability transition rate, $M_{fi}$ are the transition ...
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### Freedom in the Choice of a Beta Functions in RG

Assume we're given a certain statistical model, say the infinite range Ising model H_{N}\{\vec\sigma_{N}\}~=~ - \frac{x_{N}}{2N} \sum_{i,j =1}^{N} \sigma_{i} \sigma_{j} ...
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### Is $k_B \rightarrow 0$ the classical limit of stat. mech., as $\hbar \rightarrow 0$ is in QM?

I hear very often among my peers and seniors that just as how $\hbar\rightarrow0$ takes me to classical mechanics from quantum mechanics, $k_B\rightarrow0$ will take me to classical thermodynamics ...
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### How much energy Maxwell's demon will earn?

Suppose we have one mole of one-atom ideal gas at temperature $T$. Suppose Maxwell's daemon has separated molecules into two sections, one with speed below mean and another with speed above mean. ...
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### Why does Physica A journal contain economics papers? [closed]

Why does Physica A journal contain economics papers? Like this: Steve Keen, Russell Standish. Proﬁt maximization, industry structure, and competition: A critique of neoclassical theory. Physica A ...
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### few fermions in a harmonic trap — position density matrix from diagrammatics

I'm trying to calculate the momentum distribution of a 1D system of non-interacting identical fermions in a harmonic trap. Given Feynman's answer (from his Statistical Mechanics book) for the ...
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### Why is the transition into N proportional to N+1?

I am having trouble understanding the origin of the bosonic stimulated emission. How can I qualitatively understand why bosons Boson's attract each other into similar quantum states. The furtherst I ...
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### Ways to experimentally control the chemical potential of a solid state system

When working in the grand canonical ensemble we write the grand potential as $\Omega = \Omega (T,V,\mu)$. In this case we are taking the chemical potential $\mu$ to be an independent variable. This ...
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### Influence of choice of statistics on gas kinetics

In the derivation of distributions over energy states, a common assumption made is that under normal conditions (normal from a fluid dynamics standpoint, so > 300K typically) the energy states are ...
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### The strong Markov property of Gibbs measures in 2D Ising Model

My background is that of a mathematician. I have a question about the two Dimensional Ising Model. I think the terminology I use is similar to the physical. I'm trying to understand the following ...
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### Why can we not reduce the size of a system below the correlation length without qualitatively changing its properties?

This question is posed in the context of thermodynamics/statistical mechanics. Suppose we define the correlation length as the $\xi$ in the exponential factor $e^{-r/\xi}$ that appears in the ...
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### Partition function of bosons vs fermions

I have two atoms, both of which are either bosons or fermions, with four allowed energy states: $E_1 = 0$, $E_2 = E$, $E_3 = 2E$, with degeneracies 1, 1, 2 respectively. What's the difference between ...
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### How do I calculate the probability that the oscillator is in a certain state using partition function?

So let's say I have a single ($N=1$) quantum harmonic oscillator and the energy is determined by $E_n = (n + 1/2) \cdot \hbar \omega$ (where $n$ is the quantum number and $n$ = $0, 1, 2, \ldots$) ...
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### Velocity of real gas molecules?

It is known that the velocity of ideal gas molecules can be computed using Maxwell-Boltzmann law of distribution of molecular velocities, with average velocity given as: ...
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### General relativity and the microscopic/macroscopic distinction

Here is Wikipedia's diagram of the stress-energy tensor in general relativity: I notice that all of its elements are what would be termed "macroscopic" quantities in thermodynamics. That is, in ...
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### What does Metric Transitivity Mean?

Jaynes In his paper "Information theory and Statistical mechanics" says "Previously, one constructed a theory based on equations of motion, supplemented by additional hypothesis of ergodicity, ...
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### The analogy between temperature and imaginary time

There are many statements about the relation between time and temperature in statistical physics and quantum field theory, the basic idea is to interpret (inverse) temperature in statistics as "time" ...
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### Physics-based derivation of the formula for entropy

I am looking for a derivation of the formula $$S~=~-\Sigma_ip_i \log (p_i).$$ for entropy, from first principles. I only wish to assume the laws of physics, and without involving concepts in ...
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### Fock picture of bosonification in condensates

I want to understand how bosonification in a condensate must be interpreted in the Fock states picture Say i have uncoupled fermions in a set of states $E_1$, $E_2$ ... over the vacuum $E_0$. They ...
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### Limit of Fermi-Dirac distribution as $T$ goes to zero

Hopefully this is a simple question, I just can't seem to get my mind around it. I'm to take the limit of the Fermi-Dirac distribution for $T \rightarrow 0$. In this limit the chemical potential is ...
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### What is non-thermal plasma?

I read about non-thermal plasma, but I still have some questions: The ions and neutral particles are not in thermal equilibrium with the electron, does that mean that the overall temperature is low ...
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### The definition of entropy in quantum mechanics

I have seen entropy with several different definitions. Like Von Neumann entropy and Rényi entropy, etc. So I am curious why there are so many different definitions in quantum mechanics while only ...
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### Spin 3/2 Statistical Mechanics Problem

I am trying to solve a problem from the book 'Introductory Statistical Mechanics' (Bowley, Sanchez). The question reads: Calculate the free energy of a system of N particles, each with spin 3/2 with ...
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### An explanation for the Landauer's principle

Has anyone understood the Landauer's principle? What is the current status? In specific, is there a theoretical derivation of the Landauer's Principle?(not the heuristic one based on Salizard's ...
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### Do laws of thermodynamics have a place in Theory of Everything? [closed]

I am having a difficulty understanding why second law of thermodynamics is still a valid universally accepted concept. I understand it works on paper for describing isolated heat systems. However, I ...
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### Does high entropy means low symmetry?

According to Bogolubov postulate (various texts name it differently) in Non-equilibrium thermodynamics, the number of needed parameters to describe our system is decreasing with time, and finally at ...
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### Edwards-Anderson Hamiltonian of a Hopf link

I was calculating the Edwards-Anderson Hamiltonian of a Hopf link. A hopf link is like attachment 1. I have drawn the Seifert surface of that link. The surface is shown in attachment 2. It also ...
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### Is it possible to find the number of gas atoms/molecules in a box when the number is small?

Given very low number of particles in a system (e.g. in the 100s), is there a way to accurately measure the number of particles in the system? Assume temperature, pressure and volume is constant and ...
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### Why can $\beta$ not be linearly proportional to $T$, that is $\beta = constant \times T$?

$\beta$ in statistical mechanics is equal to $\frac{1}{k_BT}$ in in thermodynamics, but I do not understand why $\beta\propto T^{-1}$ instead of, say, $\beta\propto T$?
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### Formal demonstration that minimizing the free energy equals maximizing the entropy

I never had great intuition when it came to thermodynamic concepts and potentials even though reading a textbook and completing the exercises has never been a huge problem. In one of them, I was ...
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### Why does the law of increasing entropy, a law arising from statistics of many particles, underpin modern physics?

As far as I interpret it, the law of ever increasing entropy states that "a system will always move towards the most disordered state, never in the other direction". Now, I understand why it would ...
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### Can closed loops evade the spin-statistic theorem in 3 dimensions?

The famous spin-statistics result asserts that there are only bosons and fermions, and that they have integer and integer-and-a-half spin respectively. In two-dimensional condensed matter systems, ...
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### Simulating quantum network of harmonic oscillators

Let's say that I have a system of $n$ particles $p_1,\ldots,p_n\in\mathbb{R}^3$ (where $n$ here is on the order of 10,000). Furthermore, suppose we have a graph $G=(V,E)$ describing some network, ...
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### What is a bulk phase transition?

I have been able to google "bulk phase transition" and get plenty of results that verify that something called a bulk phase transition exists, however, I cannot seem to find a precise definition of ...
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### Why does bad smell follow people (assuming they are not the source)?

When you are sitting in a room where there is a source of bad smell, such as somebody smoking or some other source of bad smell, it is often a solution to simply move to another spot where bad smell ...
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### Chemical reaction as state transition?

When considering diffusion of chemicals, the reaction part is business of chemical kinetics, where the relevant characteristics of different substances come from collision theory together with some ...
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### How to interpret a negative failure rate?

In statistical engineering the "hazard rate" of a distribution is defined as: $$r(x)=\frac{f(x)}{1-F(x)}$$ where $f(x)$ and $F(x)$ are the PDF and CDF. Basically $r(x)$ is the odds that, having ...
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### Confusion about Free Energy and the Hamiltonian

I'm probably making a relatively basic mistake here, but I'm a bit confused about the relation between the Hamiltonian and Helmholtz free energy. From what I can see, the free energy can be written ...
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### Pair interactions on finite square lattice

I am looking for an exact or approximate solution to a statistical lattice-particle problem: Given a lattice of size $L\times L$ where $\rho\cdot L^2$ particles are randomly distributed, calculate ...