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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Plotting phase diagrams
I was reading this article on Landau theory https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/07%...
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Is flowing with RG in 1D Ising model equivalent to changing the temperature of the system
Let us consider the easiest form of the Ising Hamiltonian:
$$ \beta H(s_i; J) = -J\sum_i^N s_i s_{i+1} $$ ($\beta = 1/k_BT$ so we already defined $J = \tilde{J}/k_BT$ with $\tilde{J}$ constant).
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Vacuum chamber end plate thicknees calculation [closed]
How to calculate the thickness of the flat covers used in the cylindrical vacuum chamber.
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Why are the microstates of a harmonic oscillator considered equally probable, when the particle spends more time at locations of zero momentum?
Introductory statistical mechanics for a microcanonical ensemble (a.k.a. constant energy, no exchange of heat or work with environment) claims that all microstates in the momentum–location phase space ...
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Summing graphs in the partition function (statistical physics)
I am looking at Tong's lecture notes on statistical physics, and I wanted to understand a step in his cluster expansion better.
The goal here is to calculate the partition function in the canonical ...
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Is the concept of entropy a result of limited technology?
I know that entropy is the energy within a system that is unable to do useful work. However, there was a time in which we were unable to harness the energy of the wind or the sun, and now we can, ...
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How can I resolve this apparent paradox about the average age and the average lifetime?
I've got very much confused about distributions and am looking for quick help. Distributions are common in physics, so I humbly hope to receive an answer that will resolve my confusion.
Let's suppose ...
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Infinite number of microstates in quantum description? [duplicate]
The following argument comes from Griffiths QM second edition.
Suppose we're in a 1D infinite square well with 3 distinct particles. The total energy, then, is
$$
E = E_A + E_B + E_C = \frac{\pi^2 \...
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Holley and FKG Lattice Conditions
There's an interesting exercise (page 13, Exercise 11) in Hugo Duminil-Copin's Lectures on the Ising and Potts models on the hypercubic lattice, which states that the following 2 statements are ...
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Independence of Trace-Partition function
I am trying to calculate the partition function of the system of two completely decoupled systems. Probability-wise, the decoupled nature means that the PDF is the product of the PDF of each subsystem....
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Computation of free energy profile as a function of collective variables
I was trying to understand the following paragraph in Ref 1 about the computation of the free energy profile of collective variables:
While free energy can be expressed as the logarithm of a ...
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What is the difference between a macrostate and an ensemble?
I'm not entirely sure about the difference between a macrostate and an ensemble, though I think they are different. To me, it seems correct that both terms can be used to refer to a collection of ...
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The average of a continuous value: $\overline{O} = \int O(x) \rho(x) dx$, but coordinate invariant
I am trying to solve a Lagrange multiplier problem for the following equation
$$
L= - \int_{-\infty}^\infty \rho(x) \ln \frac{\rho(x)}{q(x)} dx + \alpha \left( 1- \int_{-\infty}^\infty \rho(x) dx \...
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If electric field inside a conductor is always zero, then why do free electrons move?
Electric field inside the conductor is zero. That means there is no electric force on electrons inside. Then how do free electrons move from atom to atom in random direction? What is the reason of ...
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Free energy of a one-dimensional harmonic oscillator
The potential energy of a one-dimensional harmonic potential can be expressed as $U(x)=\frac{1}{2}K(x-x_0)^2$, where $K$ is the force constant and $x_0$ is the equilibrium position. I'm wondering how ...
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Does Hawking radiation have a statistical physics origin like the usual derivation of Boltzmann factors?
According to Andrew Steane's Thermodynamics chapter 19 on Thermal radiation:
"The total emission from a physical object can usefully be separated in two parts: the thermal radiation and the rest. ...
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Why does the CMB have a spectrum like a black-body radiation?
Equilibrium distributions of particles (Maxwell, Boltzmann, Saha) are achieved by the particle collisions. On the other hand, photons do not interact with each other. From the introductory course in ...
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Normally distributing particle velocities
I'm trying to set up initial conditions for a collection of N particles distributed on the surface of a unit sphere so that they are uniformly distributed in position and their velocities are normally ...
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Beyond the energy equipartition, what is the uneven law?
There is a celebrated energy equipartition theorem, it works fine for many systems. But it requires the dense filling of the surface of constant energy. What if there are other conserved quantities, ...
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Phonon Distribution Factor in Phonon Emission Rate
The rate at which a phonon with wavevector $\vec{q}$ is absorbed is given by
$$\frac{1}\tau \propto n(\hbar \omega(\vec{q}))$$
This is pretty obvious to me. The more phonons there are the more often ...
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Second law of thermodynamics and Unitarity of quantum mechanics [duplicate]
From the second law of thermodynamics, we know the entropy must be increasing in an isolated system, such as our Universe. On the other hand, we have quantum mechanics which, I think, somehow tells us ...
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Measurements on macroscopic objects [duplicate]
I know that macroscopic objects undergo measurements continuously from the environment in which they are placed.
I also know that in a quantum computer one can make a measurement of only a few qubits.
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Number of microstates for a specific energy for $N$-particle system subdividing energy into small intervals
I was reading a part of Reif's Statistical Physics book that said that the number of microstates for a system containing $N$ particle and energy $E$ is proportional to $δE$ but I am confused here. if ...
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How can we explain expanding gas temperature at the microscopic level?
An insulated piston and cylinder contains an ideal gas. We pull the piston and expand the gas volume inside the cylinder.
I understand the temperature drops due to this expansion. But, how can we ...
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Gas temperature in a constant volume
An insulated container (constant volume, adiabatic) contains an ideal gas.
We open the container's hatch for a few seconds and let some particles escape from the container, then we close the hatch. We ...
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Standard deviation of kinetic energy approaches average kinetic energy
I have a simulated system of lots of particles modeled as circles moving in 2 dimensions. They bounce off each other and off of walls. Momentum and kinetic energy are conserved.
I noticed that the ...
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Adiabatic theorem with stochastic variables
Suppose a system which is driven by a stochastic complex variable $\alpha$(t). Looking at the eigensystem, both eigenvectors and eigenvalues are then stochastic variables. In my case, after building a ...
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How is electrostatic potential distributed along a circuit element?
Suppose we have a diode circuit like that:
Suppose the voltage has magnitude $\varphi$ and one end of the wire has potential zero.
How will the potential be distributed throughout the diode?
Does $\...
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Why constant voltage applied to pn-junction produces constant current throughout the junction?
Is this assumption just something that turns out to be experimentally valid or there is at least some mathematical model like Kronig-Penney + some statistical mechanics that is able to give a good ...
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Electric field on a small sphere too large
I used the Fermi energy for a given material to find the number of free electrons found on it's surface:
Where the Fermi energy E_f was given from another calculation using band theory. The solution ...
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What is wrong with my single Nose-Hoover thermostat?
I am trying to implement a single Nose-Hoover thermostat inside of my leapfrog velocity verlet algorithm in Python. This is what I have so far:
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The scaling form of the free energy density
I came across ''the finite size scaling'' while reading on the scaling hypothesis in statistical mechanics. I was wondering how can one derive a scaling function for the free energy which is dependent ...
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Why is the ideal gas law said to be classical when it assumes quantized energy states? [duplicate]
We are told in my thermodynamics class that the ideal gas law is a classical law that follows from classical mechanics. However, when we derived it as follows, we computed the partition function $Z_1$ ...
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Why isn't heat capacity defined as $(\frac{d\tau}{dQ})_V$ instead? [closed]
In my thermo notes, $C_V$ is defined in words to be the heat needed to change the temperature of the system. Intuitively, this definition "adds heat" and "measures change in temperature....
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Is the lattice spacing $a$ a dangerously irrelevant parameter?
Near a renormalization group fixed point, we can perform a scale transformation of length $L' = b^{-1} L$. In this case the relative lattice spacing should transform as $a' = b^{-1} a$. After $n$ ...
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Microstates in a quantum gas and the occupation number
The occupation number of a gas of fermions or bosons is typically defined as:
$$\bar{n} = \sum_n n\ p_n$$
where n is the number of particles in each state.
But $n$ is just a number of fermions (or ...
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Density of states of Fermi gas derivation
I'm going over this book. While deriving the gensity of states for a gas of fermions the author makes the following argument:
Remember that we are treating the
gas as having a set of states that can ...
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Integrating the phase space probability density function [closed]
If we consider a thermodyn. system a gas made out of N classical non interacting particles, with fixed energy. The appropriate ensemble to associate to this system is the Microcanonical ensemble. I ...
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Simplifying the master equation of a system
For a system described by a Hamiltonian $H= p_{x}^{2}/2m + b x$ with $b$ being a constant, the master equation for the density matrix ($\rho$) reads
$$\partial_{t} \rho(t,x) = -i [H (c), \rho (t,x)].$$...
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Is Thomas-Fermi screening the result of a Thomas-Fermi model in an external field?
On the one hand, the theory of Thomas-Fermi screening describes the response of an ideal metal (or electron gas) in the presence of an external field. In this theory, one assumes that you have a Fermi ...
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Confusion About Entropy
I have a lot of confusion around the concept of Entropy.
If we know a system's entire microstate, the number of ways it can exist in that microstate is 1. So, if we fully define our system, it will ...
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Are Distribution functions really the probability or the number of particles?
I am studying the distribution functions in statistical mechanics (Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac).
Are these distribution functions give the number of particles in an energy level ...
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The actual proof of $\delta W = P dV$
All "proofs" of $\,\delta W = P dV\,$ that I saw involve some piston and a gas cylinder. But how to prove this is true in general, that whenever (or maybe under some necessary conditions) we ...
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Entropy change in free expansion
In the textbook of thermodynamics by Zemansky, I came to free expansion. If $dQ=0$ (because there is no heat exchange between system and surroundings), entropy should be $0$ as $dS=dQ/T.$ Now, the ...
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Ising Model in 1D with long range interaction [closed]
I would like to arrive at the critical temperature for an Ising model with a long-range power-law Hamiltonian given by:
$$H = -\sum_{i=1}^{N}\sum_{j<i}\frac{J}{\lvert i - j\rvert^\alpha}\sigma_i \...
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Number of microstates at energy $E$ vs Number of microstates at energy from 0 to $E$
I'm reading this book and here the author infers that the number of microstates at energy E can be approximately equal to the number of microstates at energy from 0 to E by showing that in a concrete ...
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Confusion about average energy with boltzmann distribution
When calculating the average kinetic energy of a gas, should we use
$$ <E > = \frac{\sum_{i=0}^{\infty} E_i e^{-E_i/kT}}{\sum_{i=0}^{\infty}e^{-E_i/kT}}\approx \frac{\int_0^\infty \frac{1}{2}mv^...
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Why do heating curves have plateaus? [duplicate]
The heating curve of a solid looks something like this:
Why do the plateaus occur, various explanations online say that it is because the energy is being used to break the bonds(why is not bond ...
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Is there sufficient "content" in the field of econophysics to write a substantial undergraduate thesis/project on? [closed]
Okay, maybe the title is somewhat misleading. My university calls this a BSc Project, but it is limited to between 4000 and 6000 words, so it isn't particularly long. Anyhow, one of the projects ...
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Dimensional inconsistency in evaluating the canonical partition function
We know that canonical partition of an $N$-particle system is given as
$$Z=\!\!\!\!\!\!\!\!\!\!\!\!\sum_{\text{All possible microstates}}\!\!\!\!\!\!\!\!\!\!\!\!e^{-\beta E}=\sum_E\Omega(E)e^{-\beta E}...
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