Ensemble Definition and 125 Threads

I've recently had a look back at the review by Home and Whitaker as well as Asher Peres's book "Quantum Theory: Concepts and Methods". Both "minimal interpretation" and "ensemble interpretation" refer to categories of interpretations rather than interpretations themselves. Specifically, you can...

Hi, I've a question about the concept of ensemble is statistical physics. Take a conservative system in a given macrostate (e.g. with a given energy): there will be a number of phase space's microstates compatible with the given macrostate. If I understand it correctly, basically the...

The question is as seen below: My attempt (note that my questions are in bold below) is below. Please note that I am self-studying AM: (a) By the independence of any interval ##dt## of time and time symmetry, we expect these two answers are the same (Is there any way to make this rigorous?)...

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...

"We argue that the ψ-ontic/epistemic distinction fails to properly identify ensemble interpretations and propose a more useful definition. We then show that all ψ-ensemble interpretations which reproduce quantum mechanics violate Statistical Independence." https://arxiv.org/abs/2109.02676

Hello. I am looking for some materials related to the ensemble average. Specifically, suppose there is a function ##A(x)## satisfying a Gaussian white noise $$\left < A(x)A(x') \right > =A_0^2\exp \left ( -\frac 1 {L^2}(x-x')^2\right )$$ where the average is taken over an ensemble. Now I need...

While classical mechanics uses single action optimizing trajectory, QM can be formulated as Feynman ensemble of trajectories. As in derivation of Brownian motion, mathematically it is convenient to use nonphysical: nowhere differentiable trajectories - should it be so? Can this connection be...

I am a big fan of Ballentine's book on QM and was reading the discussions about the Ensemble Interpretation. Although, I am not an expert on these matters I reject the idea of the wave function collapse as a fundamental postulate of QM. Instead, I've come to the conclusion that we don't...

[Moderator's note: Spun off from another thread due to topic and subforum change.] I think Ballentine's interpretation is ruled out by the PBR theorem. Maybe we could discuss that?

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

Quantum mechanics is often said to be equivalent with Feynman path ensemble, which "after Wick rotation" becomes Boltzmann path ensemble, also called euclidean path integrals (popular for numerical calculations), or random walk/diffusion MERW (maximal entropy random walk). But Boltzmann path...

In the discussion of the pressure in macrocanonical ensemble, I found in textbook that: ##dW = \bar p dV## (##dW## is in fact d_bar W, yet I can't type the bar) The derivation goes like: ##\bar p = \frac{1}{Z} \sum_{r} e^{-\beta E_r} (-\frac{\partial E_r}{\partial V}) = ... = \frac{1}{\beta}...

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...

I have a problem to understand why this problem is microcanonical ensemble problem? And why entropy is calculated as S(E,N,V)=\ln \Gamma(E,N,V) When in microcanonical ensemble we spoke about energies between ##E## and ##E+\Delta E##.

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...

I don't know how to solve that integral, and to calculate the number of microstates first, then aply convolution and then integrate to find the volume of the phase space seems to be more complicated. Any clue on how to solve this? Thank you very much.

I've been reading up on the ensemble interpretation (aka statistical interpretation) of QM and it's making a bit more sense to me that it did on the onset, but I still have some questions about how it is consistent with experimental observations of various QM experiments, especially...

With the Liouville's theorem $$\frac{{d\rho }}{{dt}} = \frac{{\partial \rho }}{{\partial t}} + \sum\limits_{a = 1}^{3N} {(\frac{{\partial \rho }}{{\partial {p_a}}}\frac{{d{p_a}}}{{dt}} + \frac{{\partial \rho }}{{\partial {q_a}}}\frac{{d{q_a}}}{{dt}})} = 0$$ when we calculate the time evolution...

Smolin latest book about the quantum is quite interesting. Its called "Einstein Unfinished Revolution: Search for What Lies Beyond the Quantum" and he has a new theory or interpretation. Id like to know what you make of it. The theory is very simple. Similar views produce QM. May i know how this...

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...

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...

I know the PF rules regarding using only peer reviewed texts, but in this case, it's the only source I have access to and I'm not trying to promote the content in any way, really I'm just trying to understand if it is actually saying what it seems to be. https://arxiv.org/pdf/1303.3752...

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...

Homework Statement The fundamental equation of a system of \tidle{N} atoms each of which can exist an atomic state with energy e_u or in atomic state e_d (and in no other state) is F= - \tilde{N} k_B T \log ( e^{-\beta e_u} + e^{-\beta e_d} ) Here k_B is Boltzmann's constant \beta = 1/k_BT...

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...

The statistical ensemble interpretation (SEI) is supposed to be a minimal interpretation of QM with the smallest amount of philosophy, vagueness and controversy. Yet it turns out not to be the case. For instance Ballentine, the inventor of SEI, interprets Bell theorem as a strong evidence of...

Homework Statement Show that: d<A(q,p)>/dt=<{A,H}>, where {A,H} is a Poisson Bracket Homework Equations Liouville theorem The Attempt at a Solution <A>=Tr(Aρ)⇒d<A>/dt=Tr(Adρ/dt)=Tr(A{H,ρ}) So, in order to get the correct result, Tr(A{H,ρ}) must be equal to Tr({A,H}ρ), but I don't think I can...

I'm trying to solve a problem where I am given a few matrices and asked to determine if they could be density matrices or not and if they are if they represent pure or mixed ensembles. In the case of mixed ensembles, I should find a decomposition in terms of a sum of pure ensembles. The matrix...

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...

Please translate to English, so that it can be discussed here! I am primarily interested in how the virtual ensemble differs from an ordinary statistical ensemble, i.e., a large collection of actually identically prepared systems. The latter is the usual ensemble on which one can make...

Homework Statement I have ##C= NK_B (\frac{\epsilon}{K_B T})^{2}e^{\frac{\epsilon}{K_B T}}\frac{1}{(e^{\frac{\epsilon}{K_BT}}+1)^2} ## and need to sketch ##C## vs. ##T## Homework Equations See above The Attempt at a Solution I have ##C= NK_B (\frac{\epsilon}{K_B...

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...

The ensemble interpretation asserts that QM is only applicable to an ensemble of similarly prepared systems and has nothing to say about an individual system and in this way, it seems, it can prevent the need for introducing the concept of wave-function collapse and so it may seem that there is...

Homework Statement Confused about what a statistical ensemble actually means. Why does the ensemble have to have a uniform probability distribution at equilibrium? [If my definition of an ensemble is correct] The Attempt at a Solution This is what I understand so far: [/B] For any given...

Hi, I was studying about the statistical ensemble theory and facing some problem to understand these concepts , I have understood that the ensemble is a collection of systems which are macroscopically identical but microscopically different . In some books they are called as systems with...

In the following, I want to consider both photons in a sharply focussed, monochromatic beam of light (''type P'') and electrons in an electron beam (''type E'') on the same footing. In the following, X is either P or E. If we only concentrate on the internal degrees of freedom, both kinds of...

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...

I simulated a microcanonical ensemble of 10 ideal gas particles in one dimension and yielded the expected normal distribution of velocities. However, I still did not get how the algorithm works. The demon has non-negative energy content and the demon together with the system constitutes a closed...

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...