All Questions
Tagged with differentiation statistical-mechanics
29
questions
2
votes
1
answer
39
views
"Why is $n$ held constant when taking the time derivative in the course of the Van Kampen's system size expansion?"
I follow the notation used in "stochastic processes in physics and chemistry"(p.245) by Van Kampen.
Left hand side of master equation is
$$\frac{\partial P(n,t)}{\partial t}=\cdots.$$
We ...
- 23
5
votes
2
answers
380
views
Meaning of the differential entropy
The definition of differential (or continuous) entropy is problematic. As a matter of fact, differential entropy can be negative, can diverge and is not invariant with respect to linear ...
- 186
1
vote
1
answer
62
views
Weird derivative with respect to inverse temperature identity in Tong's statistical physics lecture notes
While reading David Tong's Statistical Physics lecture notes (https://www.damtp.cam.ac.uk/user/tong/statphys.html) I came across this weird identity in page 26 (at the end of the 1.3.4 free energy ...
- 51
0
votes
2
answers
63
views
How do extreme points work in Statistical Mechanics?
Suppose that I have an $S,V,N$ ensemble. Every variable is a function of the other variable: $U(S,V,N)$, $S(U,V,N)$, $V(S,U,N)$ and $N(S,U,V)$. The functions are everywhere differentiable. But there ...
- 1,440
2
votes
1
answer
61
views
Expanding state variables and state functions of a thermodynamic system
In this Wikipedia article under the section "Heat capacities of a homogeneous system undergoing different thermodynamic processes" there is on line that says:
$$
\delta Q=dU+pdV=\bigg(\frac{\...
- 131
6
votes
2
answers
2k
views
How to deal with differentials? [duplicate]
I am currently working on this. More specifically my question is about Problem 2.5 b). In the solution they get from
$$
Nd\mu=-SdT+VdP
$$
to
$$
N\Big(\frac{\partial\mu}{\partial N}\Big)_{T,V}=V\Big(\...
- 131
0
votes
1
answer
60
views
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 ...
- 87
0
votes
1
answer
94
views
Holding things constant in Statistical Physics for differentiation
I just want to know if the following is correct:
If one wants to verify e.g. the Maxwell relation for the ideal gas $$\left(\frac{\partial T}{\partial V}\right)_{S,N}=- \left(\frac{\partial p}{\...
- 425
2
votes
0
answers
203
views
Thermodynamics Chain Rule And Independent Variables
I was reading my textbook and I came up across the entropy $S(T,V,N)$ where temperature $T$, volume $V$, and number of particles $N$ are the independent variables. According to the chain rule the ...
- 63
0
votes
1
answer
153
views
How does one calculate partial derivatives with two constant variables in statistical mechanics
I came across this relation which I have yet to be able to prove or find proof of:
$$kT^2\left(\frac{\partial \ln\mathscr{Z}}{\partial T}\right)_{V,\mu}=\langle H\rangle-\mu\langle N\rangle$$
I was ...
- 23
1
vote
2
answers
108
views
I'm having trouble understanding exactly what $δ$ represents in thermodynamics [duplicate]
I know that $δ$ sometimes represents the Dirac delta function but in my book it states "Suppose that equilibrium has been established Then a slight change in the position of the piston should not ...
- 73
0
votes
1
answer
59
views
Functions of State and Clairaut's Theorem
In deriving the conditions for a function $f(x,y)$ to be a function of state, we end up finding out that it needs to satisfy Clairaut's Theorem:
$$
f_{xy} = f_{yx}
$$
Where
$$
df = F_1 dx + F_2dy
$$
...
- 47
1
vote
1
answer
463
views
Currently self-studying QFT and The Standard Model by Schwartz and I'm stuck at equation 1.5 in Part 1 regarding black-body radiation
So basically the equation is basically a derivation of Planck's radiation law and I can't somehow find any resources as to how he derived it by adding a derivative inside. Planck says that each mode ...
-3
votes
2
answers
118
views
Explain this equation mathematically
$$\Bigl( \frac{\partial S}{\partial T} \Bigr)_H = \Bigl( \frac{\partial S}{\partial T} \Bigr)_M + \Bigl( \frac{\partial S}{\partial M} \Bigr)_T \Bigl( \frac{\partial M}{\partial T} \Bigr)_H$$
How can ...
- 446
2
votes
3
answers
967
views
Why is this Taylor Expansion, Leading to the Boltzmann Distribution, Acceptable?
In Stephen Blundell's "Concepts in Thermal Physics" chapter 4 he derives the Boltzmann distribution. The equation that leads to the Taylor expansion is the following:
$$P_s(\epsilon) \propto ...
- 385