The study of large systems through coarse graining microscopic descriptions, providing a more detailed understanding of thermodynamics.
2
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0answers
42 views
Evolution of black holes ensemble
If the Universe contained only black holes with a certain mass and velocity distribution, how would it evolve over time? Is it enough to know the mass/velocity distribution to predict the general ...
1
vote
0answers
33 views
Maxwell-Boltzmann distribution
The short story is, that I have to calculate some transport coefficients, but using the the MB distribution as my distribution function.
What I currently need to solve is:
${{\mathcal{L}}^{\,\left( ...
5
votes
4answers
152 views
Can a single molecule have a temperature?
A show on the weather channel said that as a water molecule ascends in the atmosphere it cools. Does it make sense to talk about the temperature of a single molecule?
1
vote
1answer
43 views
How to derive the expression for Bose-Einstein distribution variance?
Can anyone point me to a derivation of this expression? $n_s$ is the number of bosons in a state.
0
votes
1answer
50 views
From Fermi-Dirac to Maxwell-Boltzmann statistics
I have a little question I can't seem to find the answer to. It is as follows:
When does Fermi-Dirac statistics reduce to Maxwell-Boltzmann statistics?
3
votes
0answers
41 views
How long would it take for a container in vacuum to leak half of its air?
Let's say I know the size of the container, size of the hole the air leaks through, pressure the air is under and temperature of the air if that helps anything. Is it possible to calculate this only ...
2
votes
0answers
29 views
Relevant operators in two dimensional O(n) models
The most general hamiltonian of a two dimensional $O(n)$ and $Z_2$ invariant statistical model can be written:
$$
H=\int d^2 x \left[\frac{\nabla \mathbf{\phi}^2}{2} + \frac{m_0^2}{2}\mathbf{\phi}^2 ...
6
votes
1answer
65 views
Dependence of chemical potential to zero point of energy
The chemical potential is defined as:
$$
\mu = -T\frac{\partial{S(N,V,E)}}{\partial{N}}
$$
It seems to me that this is completely independent of where I put the reference point of energy, because only ...
1
vote
1answer
44 views
Sackur-Tetrode equation - clarification required - problem with units
I'm a 2nd year physics undergraduate and recently I've volunteered to give a short presentation on the Sackur-Tetrode equation derivation and its use at removing the Gibbs paradox. I've looked on the ...
6
votes
1answer
87 views
Precise statement of Mermin–Wagner theorem
Roughly speaking, Mermin-Wagner theorem states that continuous symmetries cannot be spontaneously broken at finite temperature in systems with sufficiently short-range interactions in dimensions ...
1
vote
1answer
54 views
+100
Leap from photon gas energy distribution to black body radiation?
I remember considering in class in college, the case of a photon gas trapped in a d-dimensional box as a subject of interest, whose energy distribution, heat capacity, etc. should be calculated.
...
2
votes
3answers
81 views
Question about the proof that heat capacity goes to zero if temperature approaches $0K$
I don't completely understand the proof that is given for the claim that the heat capacity goes to zero, if the temperature approaches $0K$.
They do it as follows, if $C_x$ is the heat capacity where ...
2
votes
3answers
125 views
What is the general statistical definition of temperature?
Temperature in an isolated system is defined as:
$$\frac{1}{T} = -\frac{\partial{S(E,V,N)}}{\partial{E}} $$
But I wonder how one can generalize this to a random system.
Or for instance to a point in ...
0
votes
0answers
53 views
Why does the cross derivative of the partition function disappear here?
They state that the chemical potential in a canonical ensemble is given by:
$$\mu = -kT \frac{\partial{\ln Z(N,V,T)}}{\partial{N}} \tag{1}$$
But if I use the definition of chemical partial (which I ...
0
votes
0answers
26 views
Existence of Boltzmann Distribution With Constraints [closed]
I have a problem with showing the existence of Boltzmann distribution given some constraints.
Consider $p_1,...,p_n$ a Boltzmann distibution, where $p_i=\frac{\epsilon^{-\beta \cdot E_i}}{\sum_{j}^{} ...
2
votes
2answers
120 views
Why is the temperature zero in the ground state?
This is probably a simple question: I see this claims in many books, but I can't figure a reason why this is true.
So my question is why this claim is true:
"If we know that the system is in the ...
1
vote
1answer
62 views
Question about the Boltzmann distribution
In the derivation of the Boltzmann distribution they consider a system $A$, enclosed by a diathermal wall in a heat reservoir $R$. Then they calculate the probability that the system $A$ is in an ...
0
votes
0answers
29 views
Maxwell-Boltzmann distribution for transport equations
I have to calculate the transport coefficients for the Maxwell-Boltzmann distribution. But I'm not sure what distribution I have to use.
As far as I know it should not be the MB distribution for ...
3
votes
0answers
46 views
Lattice model completely constrained by boundary data
I am dealing with a lattice model that has the peculiar property that if I specify all the spins on the boundary, by local conservation laws, the whole lattice configuration (throughout the whole ...
2
votes
1answer
63 views
NP-completeness of non-planar Ising model versus polynomial time eigenvalue algorithms
From the papers by Barahona and Istrail I understand that a combinatorial approach is followed to prove the NP-completeness of non-planar Ising models. Basic idea is non-planarity here. On the other ...
1
vote
2answers
65 views
Energy dependent Maxwell-Boltzmann distribution
I'm having a bit of a problem figuring out the energy dependent Maxwell-Boltzmann distribution.
According to my book (Ashcroft & Mermin) they write the velocity dependent distribution as:
...
2
votes
2answers
57 views
What is the derivation for the exponential energy relation and where does it apply?
Very often when people state a relaxation time $\tau_\text{kin-kin}, \tau_\text{rot-kin}$,, etc. they think of a context where the energy relaxation goes as $\propto\text e^{-t/\tau}$. Related is an ...
4
votes
3answers
180 views
Definition of entropy
In physics, the word entropy has important physical implications as the amount of "disorder" of a system. In mathematics, a more abstract definition is used. The (Shannon) entropy of a variable $X$ is ...
2
votes
0answers
41 views
What is the minimum non-integer dimension for which the XY model shows a phase transition? (if well-defined)
I know that XY statistical model for $d=2$ doesn't show a regular phase transition , while the $3d$ has, I was wondering what is the behaviour for $2< d < 3$.
If it is simpler one could ...
4
votes
2answers
89 views
What would happen if energy was conserved but phase space volume wasn't? (and vice-versa)
I'm trying to understand the relationship between the two conservation laws. As I understand, Liouville's result is a weaker condition: it relies merely on the particular form assumed by Hamilton's ...
1
vote
4answers
86 views
The Preference for Low Energy States
The idea that systems will achieve the lowest energy state they can because they are more "stable" is clear enough. My question is, what causes this tendency? I've researched the question and been ...
2
votes
0answers
66 views
Ising Hamiltonian for relativistic particles
An Ising system is described by the simple Hamiltonian:
$$H = \sum\limits_{i} c_{1i} x_{i} + \sum\limits_{i,j} c_{2ij} x_i x_j \,\,\,\,\,\,\,\,\,\,(1)$$
Here the $x_i$ are spins (+1 or -1 in units ...
3
votes
4answers
209 views
If particles can find themselves spontaneously arranged, isn't entropy actually decreasing?
Take a box of gas particles. At $t = 0$, the distribution of particles is homogeneous. There is a small probability that at $t = 1$, all particles go to the left side of the box. In this case, entropy ...
1
vote
1answer
68 views
What is the interface tension between ordered and disordered phases of the Potts model?
I read in these papers(1,2) the concept of interface tension. I can't understand its definition. I can hardly imagine there is some tension in a model. Any help will be appreciated.
2
votes
1answer
41 views
Temperature of a small system
What is wrong if I define temperature of a small system (I mean, a system which has not a large number of particles) by
$$1/T = dS/dE$$
?
3
votes
1answer
129 views
Phase space in quantum mechanics and Heisenberg uncertainty principle
In my book about quantum mechanics they give a derivation that for one particle an area of $h$ in $2D$ phase space contains exactly one quantum mechanical state.
In my book about statistical physics ...
4
votes
2answers
86 views
Independent systems and Lagrangians
Definition 1:
The notion of independent systems has a precise meaning in probabilities. It states that the (joint) probability or finding the system ($S_1S_2$) in the configuration ($C_1C_2$) is ...
3
votes
0answers
32 views
Monte Carlo for Random Bond Ising ferromagnet
The set-up:
Consider the Ising model on an $L \times L$ square lattice, where the coupling of each bond is chosen to be $+J$ (ferromagnetic) with probability $(1-p)$ and $-J$ (antiferromagnetic) with ...
1
vote
3answers
75 views
Microscopic picture of an inductor
I have a good understanding of how inductors behave in electrical circuits, and a somewhat rough-and-ready understanding of how this behaviour arises from Maxwell's equations. However, what I don't ...
1
vote
0answers
32 views
Calculating the change in entropy in a melting process
I have a homework question that I'm completely stumped on and need help solving it.
I have a $50\, \mathrm{g}$ ice cube at $-15\, \mathrm{C}$ that is in a container of $200\, \mathrm{g}$ of water at ...
9
votes
3answers
254 views
Is there a way to obtain the classical partition function from the quantum partition function in the limit $h \rightarrow 0$?
One would like to motivate the classical partition function in the following way: in the limit that the spacing between the energies (generally on the order of $h$) becomes small relative to the ...
0
votes
0answers
42 views
Lambda transition data points of $\require{mhchem}\ce{^4He}$
I'm looking to get some data on the lambda transition of $\require{mhchem}\ce{^4He}$. I need the data points of the specific heat vs. temperature graph, if that makes sense.
0
votes
1answer
47 views
Number of particles in a microcanonical ensemble
Is it always assumed that, in a microcanonical ensemble, the number of particles is $N \gg 1$ ?
If no, are all the theorems related to the microcanonical description true even if the number of ...
7
votes
3answers
322 views
Why does the Boltzmann factor $e^{-E/kT}$ seem to imply that lower energies are more likely?
I'm looking for an intuitive understanding of the factor $$e^{-E/kT}$$ so often discussed. If we interpret this as a kind of probability distribution of phase space, so that $$\rho(E) = ...
6
votes
1answer
76 views
Motivation for the Deformed Nekrasov Partition Function
I have recently been doing research on the AGT Correspondence between the Nekrasov Instanton Partition Function and Louiville Conformal Blocks (http://arxiv.org/abs/0906.3219). When looking at the ...
2
votes
2answers
70 views
Error in variance
I've been exploring techniques in statistical physics, specifically applying them to spin ices. I'm in the canonical ensemble. By using the fluctuation dissipation theorem you can extract useful ...
2
votes
1answer
100 views
Energy density of a quantum mechanical ensemble
How do we determine the energy density of a given system? I have seen that the density operator
$$\rho~=~\frac{\exp(-\beta \hat{H})}{\text{tr}(\exp(-\beta \hat{H}))}.$$
What does this mean exactly ...
1
vote
1answer
67 views
Basic energy calculation for N identical spin system
We have a system that has N identical spins $n_i$, and each spin can be in state 1 or 0. The overall energy for the system is $\epsilon\sum_{i=1}^{N}n_i$.
My understanding: There is only one ...
2
votes
1answer
46 views
Bose-Einstein condensate for general interacting systems
There is Bose-Einstein condensate (BEC) for non-interacting boson systems. Can we prove the existence of BEC for interacting systems?
1
vote
0answers
30 views
Neutron scattering for a critical magnetic-ordering system : what about critical opalescence?
Liquid-gas transition critical point is believed to share the same universality class as the 3D Ising model.
We know that the liquid-gas transition is characterized by a phenomenon called critical ...
1
vote
1answer
33 views
How can I find the temperature of this system?
A system was given a small amount of thermal energy dE, and its number of states G grew by 25%. How can I find the system temperature?
The system contains gas particles, I know that $dE << ...
5
votes
1answer
78 views
Mean-field theory and spatial correlations in statistical physics
In statistical physics, mean-field theory (MFT) is often introduced by working out the Ising model and it's properties. From a spin model point of view, the mean-field approximation is given by ...
1
vote
1answer
56 views
Deriving the change in the Helmholtz free energy in the context of the free energy perturbation method
I am reading Free Energy Calculations: Theory and Applications in Chemistry and Biology by Chipot and Pohorille. At the beginning of the text (page 19, for example), the authors define the Helmholtz ...
0
votes
2answers
49 views
Has anyone studied a statistical scaling law for the universe? [closed]
How do named objects in the universe scale? Is there a predictable curve for an ordered list, say {atom, animal, planet, solar system, galaxy, etc}? Can you then use the analysis to predict when the ...
0
votes
0answers
97 views
What is the condition for getting Bose-Einstein condensation? [closed]
Consider an ideal Bose gas in three dimension with energy-momentum relation E proportional to $p^s$ with $s>0$. Find the range of $s$ for which this system may undergo a Bose-Einstein ...
