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

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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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1answer
2k views

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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1answer
922 views

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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1k views

A problem from Pathria, canonical ensemble, how to calculate $\left\langle \left(\Delta E\right)^{3}\right\rangle $

This is problem 3.18, from the book Statistical Mechanics by Pathria. Show that for a system in the canonical ensemble $$\left\langle \left(\Delta E\right)^{3}\right\rangle =k^{2}\left\{ ...
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1answer
2k views

Calculating partition function of ultra-relativistic 1D gas

This is a problem (Problem 3.16) from the book Statistical Mechanics 2nd Ed. by Pathria. In the problem I have to calculate the partition function of an ultra-relativistic 1D gas ($E_i=cp_i$) ...
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1answer
625 views

Relating the variance of the current operator to measurements

(EDIT: Thanks to Nathaniel's comments, I have altered the question to reflect the bits that I am still confused about.) This is a general conceptual question, but for definiteness' sake, imagine a ...
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1answer
300 views

Spectrum of quantum fluctuations in a harmonic oscillator

If we have a harmonic oscillator and look at it on small scale the energy is quantized and we can calculate the different eigenstates. In general the energy eigenvalues are given by $$E_n = ...
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1answer
530 views

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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936 views

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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182 views

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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1k views

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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943 views

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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157 views

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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1answer
671 views

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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1answer
1k views

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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2answers
1k views

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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2answers
671 views

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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1answer
288 views

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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1answer
281 views

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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2k views

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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73 views

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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2answers
170 views

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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4answers
441 views

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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1k views

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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3answers
2k views

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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1answer
210 views

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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1answer
307 views

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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1k views

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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333 views

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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373 views

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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1answer
157 views

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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2answers
827 views

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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2answers
167 views

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 ...
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6answers
12k views

What is a microstate, macrostate and thermodynamic probability in statistical mechanics?

Currently I am learning Maxwell-Boltsmann distribution (MBD) and in that I am learning about microstate, macrostate and thermodynamic probability (TDP). I understood the derivation of MBD but I am ...
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1k views

What are some of the best books on complex systems?

I'm rather interested in getting my feet wet at the interface of complex systems and emergence. Can anybody give me references to some good books on these topics? I'm looking for very introductory ...
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1answer
437 views

Ideal gas and diatomic gas with same temperature

If a box of ideal gas and another box of diatomic gas are in thermal equilibrium, does it mean that the average translational energy of ideal gas particle (A) is the same as that of diatomic gas ...
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2answers
187 views

Diffusion of probability amplitudes

Let's say I have a probability amplitude $\psi:\Sigma\rightarrow\mathbb{C}$ for some domain $\Sigma$ (so, $\psi$ satisfies $\int_\Sigma |\psi|^2=1$). Is there a way to use $\psi$ as initial ...
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4answers
3k views

Physics of a burning log of firewood

According to my knowledge, heat is nothing but the result of the vibrations of atoms and molecules. I guess this mean that in heating up a gas or liquid, we are increasing the rate at which the ...
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2answers
155 views

Graph Invariants and Statistical Mechanics

Many intuitive knot invariants including Jones' polynomial are inspired by statistical mechanics. Further profound connections have been explored between knot theory and statistical mechanics. I was ...
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2answers
152 views

Similarity of probability amplitude functions

Let's say I have two probability amplitude functions given by $\psi_1$ and $\psi_2$. That is, $\psi_i:\Sigma\rightarrow\mathbb{C}$ for some domain $\Sigma$ with $\int_\Sigma|\psi_i|^2=1$ for ...
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1answer
257 views

Open boundary condition and Glauber Dynamics

Warning: by background is in math, not physics. I've just recently started working with things that are close to theoretical physics. So please note that I'm still very confused by the jargon. Maybe ...
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2answers
309 views

Mathematical probabilistic interepretation of probability amplitude

As a warning, I come from an "applied math" background with next to no knowledge of physics. That said, here's my question: I'm looking at the possibility of using probability amplitude functions to ...
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3answers
752 views

Are negative temperatures typically associated with negative absolute pressures?

Negative temperatures and negative absolute pressures are both possible in physical systems. Negative temperatures arise in (for example) populations of two-state systems, which have a maximum amount ...
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4answers
204 views

What is the simplest system that has both, discontinous and continous phase transitions?

I am looking the simplest system that has both discontinous phase transition and a continous phase transition between the same phases (you can change one parameter). discontinous transition: first ...
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1answer
337 views

Black hole entropy

Bekenstein and Hawking derived the expression for black hole entropy as, $$ S_{BH}={c^3 A\over 4 G \hbar}. $$ We know from the hindsight that entropy has statistical interpretation. It is a measure ...
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1answer
184 views

Why are topological solitons present in some phases for lattice models?

Over a spatial continuum, it is easy to see why some topological solitons like vortices and monopoles have to be stable. For similar reasons, Skyrmions also have to be stable, with a conserved ...
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2answers
443 views

Micro-canonical ensemble and classical reality

I seem to find a contradiction in the notion of probability density used by Landau and the notion of micro-canonical ensemble. To see this, take an isolated classical system and we know ...
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1answer
1k views

Why is the partition function called ''partition function''?

The partition function plays a central role in statistical mechanics. But why is it called ''partition function''?
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481 views

State-dependent diffusions: Fick's law vs. Fokker-Planck's, which and why?

Consider a "state-dependent diffusion": a diffusion process for which the diffusion coefficient $D(x)$ depends on the (stochastic) state $x$ of the system. (An example is provided by the diffusion of ...
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4answers
6k views

What are distinguishable and indistinguishable particles in statistical mechanics?

What are distinguishable and indistinguishable particles in statistical mechanics? While learning different distributions in statistical mechanics I came across this doubt; Maxwell-Boltzmann ...