What is Canonical ensemble: Definition and 58 Discussions
In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat bath, so that the states of the system will differ in total energy.
The principal thermodynamic variable of the canonical ensemble, determining the probability distribution of states, is the absolute temperature (symbol: T). The ensemble typically also depends on mechanical variables such as the number of particles in the system (symbol: N) and the system's volume (symbol: V), each of which influence the nature of the system's internal states. An ensemble with these three parameters is sometimes called the NVT ensemble.
The canonical ensemble assigns a probability P to each distinct microstate given by the following exponential:
P
=
e
(
F
−
E
)
/
(
k
T
)
,
{\displaystyle P=e^{(F-E)/(kT)},}
where E is the total energy of the microstate, and k is Boltzmann's constant.
The number F is the free energy (specifically, the Helmholtz free energy) and is a constant for the ensemble. However, the probabilities and F will vary if different N, V, T are selected. The free energy F serves two roles: first, it provides a normalization factor for the probability distribution (the probabilities, over the complete set of microstates, must add up to one); second, many important ensemble averages can be directly calculated from the function F(N, V, T).
An alternative but equivalent formulation for the same concept writes the probability as
P
=
1
Z
e
−
E
/
(
k
T
)
,
{\displaystyle \textstyle P={\frac {1}{Z}}e^{-E/(kT)},}
using the canonical partition function
Z
=
e
−
F
/
(
k
T
)
{\displaystyle \textstyle Z=e^{-F/(kT)}}
rather than the free energy. The equations below (in terms of free energy) may be restated in terms of the canonical partition function by simple mathematical manipulations.
Historically, the canonical ensemble was first described by Boltzmann (who called it a holode) in 1884 in a relatively unknown paper. It was later reformulated and extensively investigated by Gibbs in 1902.
I tried to show this equality by explicitly determining what
$$ \overline{(\Delta \eta)^2} $$
is, but I got a totally different answer for some reason, here is my attempt to solve it, what did I miss?
My main question here is about how we actually justify, hopefully fairly rigorously, the steps leading towards converting the sum to an integral.
My work is below:
If we consider the canonical ensemble then, after tracing over the corresponding exponential we get:
$$Z = \sum_{n=0}^\infty...
Canonical ensemble can be used to derive probability distribution for the internal energy of the closed system at constant volume ##V## and number of particles ##N## in thermal contact with the reservoir.
Also, it is stated that the temperature of both system and reservoir is the same, i.e...
Canonical ensemble is the statistical ensemble which is applicable for the closed system in contact with the reservoir at constant temperature ##T##. Canonical ensemble is characterized by the three fixed variables; number of particles ##N##, volume ##V## and temperature ##T##.
What is said is...
One of the common derivations of the canonical ensemble goes as follows:
Assume there is a system of interest in the contact with heat reservoir which together form an isolated system. Heat can be exchanged between the system and reservoir until thermal equilibrium is established and both are...
https://scholar.harvard.edu/files/schwartz/files/7-ensembles.pdf
https://mcgreevy.physics.ucsd.edu/s12/lecture-notes/chapter06.pdf
On page 3 of both the notes above, the author merely claims that $$P \propto \Omega_{\text{reservoir}}$$
But isn't $$P \propto...
I'm trying to sort out how the microcanonical picture is connected to the canonical and the grand canonical.
If I consider a Helium gas, not necessarily with low density, in an isolated container (fixed energy and particle number) I can use the microcanonical ensemble to arrive at the...
In addition to the homework statement and considering only the case where ##U= constant## and ##N = large## : Can we also consider the definition of chemical potential ##\mu## and temperature ##T## as in equations ##(1)## and ##(2)##, and use them in the grand partition function?
More...
Hi everyone,
this is my first message after presentation so please be merciful if the notation is somewhat messy. Here's my attempt at a solution:
As for points 1) and 2) I used the definition of partition function
$$Z = \frac{1}{h^{3N}} \int e^{-\beta \mathcal{H}} d^3p d^3q$$
and the fact that...
Hi all, I am slightly confused with regard to some ideas related to the GCE and CE. Assistance is greatly appreciated.
Since the GCE's partition function is different from that of the CE's, are all state variables that are derived from the their respective partition functions still equal in...
Greetings,
I am having a hard time in understanding intuitively how pressure does not automatically stay constant in a canonical ensemble (=NVT ensemble).
Pressure in a closed system is the average force of particles hitting against the wall of said system. The obvious way to manipulate...
Homework Statement
Hi
I am looking at the question attached.
Parts c and d, see below
Homework EquationsThe Attempt at a SolutionFirst of all showing that ##<N> ## and ##<n_r>## agree
I have ##Z=\Pi_r z_r ##, where ##Z## here denotes the grand canonical ensemble.
So therefore we have ##...
Homework Statement
question attached.
My question is just about the size of the limit, how do you know whether to expand out the exponential or not (parts 2) and 4))
Homework Equations
for small ##x## we can expand out ##e^{x} ## via taylor series.
The Attempt at a Solution
Solutions...
Question
Form the canoncial partition using the following conditions:
2 N-particles long strands can join each other at the i-th particle to form a double helix chain.
Otherwise, the i-th particle of each strand can also be left unattached, leaving the chain "open"
An "open" link gives the...
Homework Statement
Consider a gas sufficiently diluted containing N identical molecules of mass m in a box of dimensions Lx, Ly, Lz.
Calculate the probability of finding the molecules in any of their quantum states.
Calculate the energy of each quantum state εr, as a function of the quantum...
So the pressure for a canonical ensemble is:
P = kbT dZ/dV
P = pressure
P = -∑pi dEi/dV
Z = ∑e-βEi
pi is the probability of being in microstate i
Ei is the energy of state i
β = 1/kbT
<E> = U = average energy
U = -1/Z dZ/dβ = -d(Ln(Z))/dβ
How can the pressure (given above) be derived in...
Homework Statement
The probability that the system has the energy ε i.e.P(ε).
The system could have any energy between 0 and E.
So, P(ε) = 1/(no. of possible systems with different energies)
I cannot understand how P (ε) is related to the no. of possible microstates the reservior could have...
I'm interested in an apparent inconsistency with the result for negative temperatures for a spin 1 system of N particles.
The partition function of such a system is
\begin{equation}
Z=(1+2\cosh(\beta \,\epsilon))^{N}
\end{equation}
where each particle can be in one of three energy states...
Homework Statement
Consider an ensamble of particles that can be only in two states with the difference ##\delta## in energy, and take the ground state energy to be zero. Is it possible to find the particle in the excited state if ##k_BT=\delta/2##, i.e. if the thermal energy is lower than the...
Homework Statement
Hi
I am looking at the attached extract from David Tong's lecure notes on statistical phsyics
So we have a canonical ensemble system ##S##, and the idea is that we take ##W>>1## copies of the system ##S##, and the copies of ##W## taken together then can be treated as a...
Hi.
In some statistical approaches (e.g. canonical ensemble), the particles of an ideal gas are non-interacting. Still, it's possible to derive the ideal gas law and other thermodynamic relations.
Wikipedia gives an equation for the speed of sound in an ideal gas. How can there be waves in a...
Homework Statement
Consider a system with N sites and N particles with magnetic moment m. Each site can be in one of three states: empty with energy 0, occupied by one particle with energy 0 (in the absent of magnetic field) or occupied by two particles with anti parallel moments and energy ε...
I must be missing some point with regards to the canonical Distribution. Let us imagine I have a closed (to energy and matter) box full of ideal gas at temperature T. The total energy in the box equals hence
E=3N2kT
, where N is the number of molecules, k Boltzmann's connstant.Next, I allow the...
Hi, I've been looking at working with the canonical ensemble and getting the probabilities of a system being at a certain energy. For reference, I am following something of the form given under 'Canonical Ensemble' in this article...
A system is in contact with a reservoir at a specific temperature. The macrostate of the system is specified by the triple (N,V,T) viz., particle number, volume and temperature.
The canonical ensemble can be used to analyze the situation. In the canonical ensemble, the system can exchange...
Hello everybody :D
My question is: given the distribution of the canonical ensemble, how do we get the helmoltz free energy?
I think we can't use A = U-TS because we don't know how to write S. So what's the solution? Thanks
Hello!
Dr. David Tong, in his statistical physics notes, derives the Boltzmann distribution in the following manner.
He considers a system (say A) in contact with a heat reservoir (say R) that is at a temperature T. He then writes that the number of microstates of the combined system (A and R)...
Suppose that we have a system of particles (I am assuming a general system), and I want to find the ground state energy ##E_0##. We know that we can consider our system by canonical ensemble formalism OR by grand canonical ensemble so that ##H_G=H-\mu N## (in which ##H## is the Hamiltonian in...
Homework Statement
We have a quantum rotor in two dimensions with a Hamiltonian given by \hat{H}=-\dfrac{\hbar^2}{2I}\dfrac{d^2}{d\theta^2} . Write an expression for the density matrix \rho_ {\theta' \theta}=\langle \theta' | \hat{\rho} | \theta \rangle
Homework Equations...
I have a problem in understanding the quantum operators in grand canonical ensemble.
The grand partition function is the trace of the operator: e^{\beta(\mu N-H)} (N is the operator Number of particle)
and the trace is taken on the extended phase space:
\Gamma_{es}= \Gamma_1 \times \Gamma_2...
Homework Statement
Statistical Mechanics by Pathria. Problem 3.1
Homework Equations
(1)
<(△nr)2>=<nr2>-<nr>2=(wrd/dwr)(wrd/dwr)lnΓ, for all wr=1
How to derive above equation from these equations?
<nr>=wrd/dwr(lnΓ), for all wr=1
<nr2>=(1/Γ)(wrd/dwr)(wrd/dwr)Γ, for all wr=1
(2)
Also, if you...
I have a short question which I have been discussing with a fellow student and a professor. The question (which is not a homework question!), is as follows:
If you shift all the energies E_i \to E_i + E_0 (thus also shifting the mean energy U \to U + E_0), does the entropy of the system remain...
I have a question that has puzzled me during the last couple of days. Suppose that there is a system (e.g. a small box filled with gas) that is connected to a heat bath (much larger than the system) at a constant temperature T. The studied system and the heat bath are thought to be isolated from...
Homework Statement
Part (a): Using grand canonical distribution, show ideal gas law ##P = nkT## holds, where ##n = \frac{\overline{N}}{V}##.
Part (b): Find chemical potential of diatomic classical ideal gas in terms of ##P## and ##T##. The rotational levels are excited, but not the...
Hi guys :cry:,
I regard an ideal diatomic gas which is in a Volume x,z and got a angular \phi:
0 \le x \le L, ~~~~~~~~~~~~ 0 \le z \le \infty, ~~~~~~~~~~~~0 \le \phi \le 2 \pi
The hamiltonian for the single particle is:
~~~~~~H= \frac{p^{2}_{z} + p^{2}_{x} }{2M} +...
Hello guys,
I would really need some help on the following problem.
Consider a non-interacting & non-relativistic bosonic field at finite temperature. We are all aware of the fact that such a statistical system is well described by the grand-canonical ensemble in the limit N→∞. However...
Homework Statement
I have the equation
Z=1/N!h3N∫∫d3qid3pie-βH(q,p)
How can I get the entropy from this equation assuming a classical gas of N identical, noninteracting atoms inside a volume V in equilibrium at T where it has an internal degree of freedom with energies 0 and ε
What...
In Statistical Mechanics, the key step in the derivation of the Canonical Ensemble is that the probability of S being in the m-th state, P_m , is proportional to the corresponding number of microstates available to the reservoir when S is in the m-th state. That is
P_m=c\Omega(E_0-E_m),
where...
Hello,
The entropy of the Grand Canonical Ensemble (GCE) is:
S = KB ln ZG + (\bar{E}/T) - μo\bar{N}/T
Helmholtz function is:
F = \bar{E} - TS = \bar{E} - TKB ln ZG - \bar{E} + μo\bar{N}
= -TKB ln ZG + μo\bar{N}
But
\partialF/\partialT = -S (From thermodynamics).
Then...
Hey,
Here is the problem:
The method by which we solve is by Langrange Multipliers, and so I believe I found the derivative of f with respects to P(i) but I have two quantities I'm sure what they equal:
Summations over i=1 to N : Ʃln(P(i)) and ƩE(i)
Thanks for any help,
S
Hello,
I was investigating a system with N indistinguishable particles, each of which can have an energy \pm \epsilon, and using the grand canonical ensemble, i.e. \Xi = \sum_{N=0}^{\infty} e^{\beta \mu N} Z_N.
But my entropy formula is S = \left( \textrm{a couple of $\sim N $ positive...
Can somebody explain to me the differences between the ensembles, and how does this differences refer to experiment?
I know that:
Microcanonical ensemble is a concept used to describe the thermodynamic properties of an isolated system. Possible states of the system have the same energy and...
Homework Statement
I'm looking for a closed form expression for the partition function Z using the Canonical Ensemble
Homework Equations
epsilon_j - epsilon_j-1 = delta e
Z = Sum notation(j=0...N) e^(-beta*j*delta e)
beta = 1/(k_B*T)
t = (k_B*T)/delta e
N is the number of excited...
Can we take the grand canonical ensemble and then switch the roles of the thermodynamic conjugate variable pair (P, V) making P (pressure) the parameter and V (volume) the variable and allowing it to fluctuate in the system. The macrostate would then be defined by the pressure temperature and...
Hi folks,
since the volume V is fixed in a canonical ensemble I'm a bit confused about the fact, that the pressure is calculated as the derivation of the internal energy U with respect to the volume V.
Sure, P = dU/dV comes from dU = dQ + dW = tdS - pdV + ... But what does it mean to derivate...
Homework Statement
In deriving <E2>-<E>2
starting from <E>=U=sum(Eiexp(-beta Ei))/sum(exp(-beta Ei). the taking derivative of U with respect to beta, the book always notes E (thus Volume) is held constant. what i am trying to do is taking the derivative of U with respect to beta or T...
In derivation of probability of system at energy E with canonical ensemble, one assumes that the probability of system in a microstate Ei is proportional to the multiplicity of reservoir. Is this probability the conditional probability by knowing that system is at energy Ei with knowing it is at...
Homework Statement
Some systems are adequately described by a one-dimensional potential in the form of an asymmetric double well. To good accuracy each can assumed to be harmonic with potential energies:
V_L(x)=\frac{1}{2}k_Lx^2, V_R(x)=\epsilon+\frac{1}{2}k_R(x-a)^2
Here, \epsilon=V_R(a)>0. N...