Canonical ensemble Definition and 50 Threads

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...

I try involving differentiating ^2 but I get an expression of different proportionality.

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...

Where does the volume even come from? Any help would be appreciated!

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...

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...

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...

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...

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...

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...

Is it true that partition function for canonical ensemble is Z=\sum_{\{states j\}}e^{-\beta E_j}=e^{-\beta F} where F is Helmholtz free energy?

Homework Statement In the canonical ensemble, the probability that a system is in state r is given by P_i = \frac{g_i \exp (-\beta E_i)}{\sum_i g_i \exp( -\beta E_i)} where g_i is the multiplicity of state i. This is confusing me because I thought P_i = \frac{g_i}{\sum_i g_i} = states...

Homework Statement Grand Canonical ensemble problem- A surface of N sites that can have 0,1,2 atoms. It costs no energy to adsorb an atom. Grand canonical problem therefore in contact with particle reservoir. Assume \mu chem potentiol and temp T. What is probability for site to be empty,1,or 2...

I've been studying and thinking about statistical physics for a couple days now... and what bothers me is the grand canonical partition function. Namely that for a system with fixed chemical potential and energy \epsilon_i the probability of having N_i particle in that state is proportional to...

Hey folks, I really need help setting up a canonical ensemble. I am using the einstein model and have an energy: E_\nu=A + \sum_{i=1}^{2N} \hbar\omega n_i+\sum_{j=1}^{N} \hbarh\omega n_j where A is some constant. Now, I need to build the partition function which I 'think' looks like...

Hi there, I have a system with the following energy using the einstein model: E_\nu=\sum_{i=1}^{2N} h\omega n_i+\sum_{j=1}^{N} h\omega n_j I need to set up a canonical ensemble for this. How would I write the partition function please?

I am taking a statictical mechanics course, and one thing bothers me. I am not sure when we should use the normal partition function (Z) and when the grand partition function (twisty Z). In particular, why can we not use grand partition function when we are considering the following system...