Qm Many Body And Equilibrium Models — Examples

Adopt P1, P2, P4 and P7 in the readings of Assigned sharp observables and the outcome law of the framework and Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model, and work in the finite-mode realization of the declared fixed-$N$ model of The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable with: one-particle space $K={\mathbb{C}}^2$, declared sites $\mathrm{\mid }1⟩,\mathrm{\mid }2⟩$ an orthonormal basis of $K$, particle number $N=1$, single site-occupation sector, and the declared one-particle operator (the bond Hamiltonian) $h:=-t(T_{\mathrm{\mid }1⟩,\mathrm{\mid }2⟩}+T_{\mathrm{\mid }2⟩,\mathrm{\mid }1⟩}),t>0\text{ in joules}\mathrm{.}$ Adopt the canonical equilibrium postulate The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature at assigned temperature $T>0$ and inverse temperature $\beta =1\mathrm{/}(k_{B}T)$, and read $\epsilon :=2t$ as the tunnelling splitting. Then:

(a) $h$ has the orthonormal eigenbasis $u_{-}:=\frac{\mathrm{\mid }1⟩+\mathrm{\mid }2⟩}{\sqrt{2}},u_{+}:=\frac{\mathrm{\mid }1⟩-\mathrm{\mid }2⟩}{\sqrt{2}},h{u}_{\mp }=\mp t{u}_{\mp },$ so its eigenvalues are $-t$ and $+t$; equivalently $h+t\mathrm{i}\mathrm{d}$ is the two-level Hamiltonian of Closed formulas for the two-level canonical weights, mean energy, entropy and free energy with levels $0$ and $\epsilon =2t$ and eigenvectors $u_{-},u_{+}$;

(b) the canonical preparation of $h+t\mathrm{i}\mathrm{d}$ and hence of $h$ is ${\rho }_{\beta }=\frac{T_{u_{-},u_{-}}+e^{-2\beta t}{T}_{u_{+},u_{+}}}{1+e^{-2\beta t}},$ with shifted partition sum $Z_0=1+e^{-2\beta t}$ and physical partition trace $Z_h=\mathrm{tr}{e}^{-\beta h}=e^{\beta t}{Z}_0$;

(c) the exact means, entropy and free energy are $⟨h{⟩}_{\beta }=\frac{2t}{e^{2\beta t}+1}-t=-t\tanh (\beta t),s({\rho }_{\beta })=\log (1+e^{-2\beta t})+\frac{2\beta t{e}^{-2\beta t}}{1+e^{-2\beta t}},F=-\frac{1}{\beta }\log (1+e^{-2\beta t})-t,$ and for every $\beta >0$ one has $-t<⟨h{⟩}_{\beta }<0$ and $0\le s({\rho }_{\beta })<\log 2$;

(d) scope: the computation is an exact statement about the declared two-dimensional bond model at the assigned temperature; it identifies no physical double-well or molecular system, asserts no measured splitting and no transition, and the temperature assignment is the tested identification of P7.

• The shift by $t\mathrm{i}\mathrm{d}$. Adding a constant to the Hamiltonian multiplies $Z$ by an exponential and leaves the canonical preparation and the entropy unchanged while shifting the free energy by the same constant; the example uses this to instantiate the supplied two-level formulas, whose levels are $0$ and $\epsilon$.
• Low- and high-temperature values. The formulas give $s\to 0$ and $⟨h{⟩}_{\beta }\to -t$ as the exponential factor is suppressed, and $s\to \log 2$, $⟨h{⟩}_{\beta }\to 0$ as it approaches one; the example states the exact formulas and the bounds, not a limit theorem.

Adopt P1, P2 and P6 in the readings of The declared finite-mode model and its occupation readouts and Assigned sharp observables and the outcome law of the framework, work in the finite-mode realization of the declared fixed-$N$ model of The declared fixed-N many-body model: species sector, one-particle data, interaction and the sector energy observable, and adopt the canonical equilibrium postulate The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature at assigned temperature $T>0$ with $\beta =1\mathrm{/}(k_{B}T)$. Fix $\epsilon >0$ in joules.

Fermionic model. One-particle space $K={\mathbb{C}}^3$ with modes $f_1,f_2,f_3$, fermionic statistics $\sigma =-$, $N=2$, zero interaction $W=0$, and the declared one-particle operator $h:=\epsilon (0\cdot T_{f_1,f_1}+1\cdot T_{f_2,f_2}+2\cdot T_{f_3,f_3})$.

Bosonic model. One-particle space $K={\mathbb{C}}^2$ with modes $g_1,g_2$, bosonic statistics $\sigma =+$, $N=2$, zero interaction $W=0$, and $h^{\mathrm{\prime }}:=\epsilon (0\cdot T_{g_1,g_1}+1\cdot T_{g_2,g_2})$.

Then:

(a) in the fermionic model the antisymmetric sector has orthonormal eigenbasis $e_{\{1,2\}}^{\wedge },e_{\{1,3\}}^{\wedge },e_{\{2,3\}}^{\wedge }$ with energies $\epsilon ,2\epsilon ,3\epsilon$; the partition sum and the exact occupation means are $Z_{\mathrm{F}}=e^{-\beta \epsilon }+e^{-2\beta \epsilon }+e^{-3\beta \epsilon },$ $⟨N_1{⟩}_{\beta }=\frac{e^{-\beta \epsilon }+e^{-2\beta \epsilon }}{Z_{\mathrm{F}}},⟨N_2{⟩}_{\beta }=\frac{e^{-\beta \epsilon }+e^{-3\beta \epsilon }}{Z_{\mathrm{F}}},⟨N_3{⟩}_{\beta }=\frac{e^{-2\beta \epsilon }+e^{-3\beta \epsilon }}{Z_{\mathrm{F}}},$ with $⟨N_1+N_2+N_3{⟩}_{\beta }=2$ and ground energy $\epsilon$ (configuration $\{1,2\}$);

(b) in the bosonic model the symmetric sector has orthogonal eigenbasis $m_{(2,0)},m_{(1,1)},m_{(0,2)}$ with energies $0,\epsilon ,2\epsilon$; the partition sum and the exact occupation means are $Z_{\mathrm{B}}=1+e^{-\beta \epsilon }+e^{-2\beta \epsilon },⟨N_1{⟩}_{\beta }=\frac{2+e^{-\beta \epsilon }}{Z_{\mathrm{B}}},⟨N_2{⟩}_{\beta }=\frac{e^{-\beta \epsilon }+2e^{-2\beta \epsilon }}{Z_{\mathrm{B}}},$ with $⟨N_1+N_2{⟩}_{\beta }=2$ and ground energy $0=N{\epsilon }_1$;

(c) scope: the computations are exact statements about the declared finite-mode models at the assigned temperature; they assert no physical species, no measured occupation, no thermalisation and no thermodynamic limit.

• Dimension counts. The fermionic sector is three-dimensional by exclusion and the bosonic sector is three-dimensional by the configuration count; the two examples are therefore directly comparable finite checks of the diagonalisation theorem at different statistics.
• Exactness. The formulas are exact rational functions of $e^{-\beta \epsilon }$; no expansion in $\epsilon \mathrm{/}(k_{B}T)$ is used, and the dilute-limit approximations are treated separately on this page.

Adopt the mode models and thermal readings of Thermal occupation means of the declared free-mode models: Bose and Fermi factors, product convergence and dilute remainders at assigned temperature $T>0$ with $\beta =1\mathrm{/}(k_{B}T)$. For a canonical single mode of energy $\epsilon >0$, put $x:=\beta \epsilon >0$ and $q:=e^{-x}\in (0,1)$. In the grand-canonical reading put $x:=\beta (\epsilon -\mu )$ and $q:=e^{-x}>0$; the bosonic formulas below apply only when $\mu <\epsilon$ (so $q<1$), while the fermionic formulas hold for every $\mu \in \mathbb{R}$. Then:

(a) the exact single-mode means and their dilute identities are $⟨N{⟩}_{\mathrm{B}}=\frac{q}{1-q}=q+\frac{q^2}{1-q},⟨N{⟩}_{\mathrm{F}}=\frac{q}{1+q}=q-\frac{q^2}{1+q},$ so the bosonic mean exceeds $q$ by $q^2\mathrm{/}(1-q)$ and the fermionic mean falls below $q$ by $q^2\mathrm{/}(1+q)$ exactly;

(b) at $q=1\mathrm{/}10$ the exact means are $⟨N{⟩}_{\mathrm{B}}=1\mathrm{/}9$ and $⟨N{⟩}_{\mathrm{F}}=1\mathrm{/}11$, and the exact identities $1\mathrm{/}9=1\mathrm{/}10+1\mathrm{/}90$ and $1\mathrm{/}11=1\mathrm{/}10-1\mathrm{/}110$ verify the two remainder formulas;

(c) for the free-mode energies ${\epsilon }_j=j\epsilon$, $j\ge 1$, the sums ${\sum }_j{e}^{-\beta {\epsilon }_j}=e^{-\beta \epsilon }\mathrm{/}(1-e^{-\beta \epsilon })$ converge, so the bosonic and fermionic product partitions are finite (bosonic case with $\mu <\epsilon$); for the constant energies ${\epsilon }_j\equiv \epsilon$ the sum ${\sum }_j{e}^{-\beta {\epsilon }_j}=+\mathrm{\infty }$ diverges, so both product partitions diverge;

(d) scope: the identities and the check are exact statements about the declared mode models at the assigned temperature and, where stated, chemical potential; no measured occupation, dilute gas, thermodynamic limit or condensation is asserted.

• What the check tests. The check confirms the two remainder identities in exact arithmetic at a fixed value of $q$; it is a finite verification of algebraic identities, not an empirical test and not a statement about any gas.
• Two ways to fail convergence. Constant free-mode energies have divergent product partition because infinitely many modes share the same positive weight; linearly increasing energies have geometrically decaying weights and finite product partition. The bosonic criterion additionally requires the chemical potential below the ground-mode energy.

Adopt P1, P2, P4 and P7 in the readings of Assigned energy observable, unitary flow and the two pictures of autonomous evolution, Assigned sharp observables and the outcome law of the framework and Entropy, the Boltzmann constant and the free-energy functional of the declared finite-sector model, work in the two-site bond model of The two-site bond model: exact canonical weights, mean energy, entropy and free energy of one particle on two sites with coupling $t>0$, sites $\mathrm{\mid }1⟩,\mathrm{\mid }2⟩$, bond Hamiltonian $h=-t(T_{\mathrm{\mid }1⟩,\mathrm{\mid }2⟩}+T_{\mathrm{\mid }2⟩,\mathrm{\mid }1⟩})$ and eigenbasis $u_{-}=(\mathrm{\mid }1⟩+\mathrm{\mid }2⟩)\mathrm{/}\sqrt{2}$, $u_{+}=(\mathrm{\mid }1⟩-\mathrm{\mid }2⟩)\mathrm{/}\sqrt{2}$ with $h{u}_{\mp }=\mp t{u}_{\mp }$, and adopt the canonical equilibrium postulate The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature at assigned temperature $T>0$, $\beta =1\mathrm{/}(k_{B}T)$. Consider the hypothetical declared preparation $\rho (0):=T_{\mathrm{\mid }1⟩,\mathrm{\mid }1⟩}$ (the particle localised at site $1$) evolving by $\rho (\tau )=U(\tau )\rho (0)U(\tau {)}^{\ast }$ with $U(\tau )=\exp (-i\tau h\mathrm{/}\mathrm{\hslash })$. Then:

(a) $\rho (0)\mathrm{\ne }{\rho }_{\beta }$, and for every $\tau \in \mathbb{R}$ one has $\rho (\tau )\mathrm{\ne }{\rho }_{\beta }$: the unitary evolution of the model never produces the canonical preparation from this preparation;

(b) the site-$1$ occupation readout $(P_1,1)$ with $P_1:=T_{\mathrm{\mid }1⟩,\mathrm{\mid }1⟩}$, assigned in the dimensionless particle-count unit, has the exact mean $⟨P_1⟩(\tau )={\cos }^2\text{\hspace{0.17em}⁣}\left(\frac{t\tau }{\mathrm{\hslash }}\right),$ and $\mathrm{tr}({\rho }_{\beta }{P}_1)=1\mathrm{/}2$; in particular $⟨P_1⟩(0)=1\mathrm{\ne }1\mathrm{/}2$, so at the initial time the readout does not take its canonical mean;

(c) the energy readout $h$ has the constant mean $\mathrm{tr}(\rho (\tau )h)=0$ for every $\tau$, while its canonical mean is $\mathrm{tr}({\rho }_{\beta }h)=-t\tanh (\beta t)<0$; hence the energy readout never takes its canonical thermal value;

(d) scope and interpretation: the scenario is a hypothetical, exactly-solved model deduction in one declared preparation; it shows that agreement or disagreement of individual readout means with their canonical values is not convergence of the preparation to the canonical one, and that the model's P4 dynamics does not produce the canonical ensemble. It asserts no observed equilibration or non-equilibration of a sample, no measured occupation or energy, and no claim about environments or open systems.

• What the scenario shows. The preparation never equals the canonical one, yet the site-occupation readout has the same mean $1\mathrm{/}2$ as the canonical preparation at the times when ${\cos }^2(t\tau \mathrm{/}\mathrm{\hslash })=1\mathrm{/}2$; the energy readout meanwhile never takes its canonical mean. Readout coincidence is therefore not state convergence, and different readouts behave differently under the same unitary flow.
• Hypothetical status. No apparatus, sampling or detection model is specified; the scenario is a conditional statement about the declared model, and any comparison with an experiment would require the declared preparation, isolation, detection and sampling models of P7.

The companion items contain three conditional examples and one hypothetical scenario in declared finite-mode or countable-mode models:

• The two-site bond model: exact canonical weights, mean energy, entropy and free energy of one particle on two sites solves the one-particle two-site bond model exactly: the tunnelling-split eigenbasis, the canonical weights and the exact mean energy, entropy and free energy;
• Finite-mode non-interacting thermal ensembles: two fermions in three modes and two bosons in two modes evaluates the canonical partition sums and occupation means of two fermions in three modes and two bosons in two modes, with the total-occupation identity and ground energies checked against the page's diagonalisation theorem;
• Free-mode statistics at low occupation: exact dilute identities, a finite arithmetic check and two convergence examples verifies the exact Bose and Fermi dilute identities, evaluated in exact rational arithmetic at $q=1\mathrm{/}10$, and contrasts the finite product partition of linearly spaced mode energies with the divergent product of constant energies;
• A site-localised preparation of the two-site bond model never becomes canonical, and its readouts need not take their thermal means shows, in one declared hypothetical preparation of the bond model, that the unitary evolution never produces the canonical preparation, that the site-occupation readout takes the canonical value $1\mathrm{/}2$ at some times while differing from it at others, and that the energy readout never takes its canonical value.

None of these items contains measured data, a measured uncertainty, a calibration, a sampling model or a detector model; all numbers are exact model values in the declared finite-mode or countable-mode idealizations, and the assignment of a physical system to any of these models is a tested identification under P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). In particular:

• the bond model is not any particular molecule, and its splitting $2t$ is an assigned parameter, not a measured spectroscopic value;
• the finite-mode examples are not ideal or interacting gases; they make no statement about a thermodynamic limit, pressure, condensation or a continuum of modes;
• the comparison of readout means with canonical values in the thought experiment is a statement about the declared preparation and the exact unitary flow; it is not evidence for or against thermalisation in nature, and the canonical ensemble remains the adopted assumption of The canonical equilibrium-ensemble assumption: the Gibbs preparation of a declared model at assigned temperature;
• the thought experiment uses the no-unitary-production conclusion (c) of Unitary evolution preserves entropy and cannot produce the canonical preparation from a different one. The separate entropy-conservation conclusion (a) of that theorem concerns the same finite closed-system class but is not a premise of the readout calculation; neither conclusion forbids or asserts open-system or environment-induced behaviour.

The cited treatments supply the model formulas used here at the recorded locators; they do not supply measured values for these particular finite-mode or countable-mode models, and no source is used as an empirical premise on this page. Where a comparison with data would be attempted, the preparation, isolation, detection and sampling models plus the uncertainty of every assigned parameter would have to be declared and carried through, as P7 requires.