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Unpinned translation zero-mode divergence

Physical scope

Finite periodic harmonic chain with no pinning; conditional on ACω for the declared Euclidean Lebesgue reference, the continuous translation zero mode makes the canonical partition function diverge.

Example

Assume ${A C}_{ω}$ as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ) for the Euclidean Lebesgue measure of the phase-space boxes below. Take $N \geq 3$ resolved displacement coordinates $q_{i} \in R$ in metres and momenta $p_{i} \in R$ in kg m s $^{- 1}$ on a periodic chain, with $q_{N + 1} = q_{1}$ , mass $m > 0$ kg, spring stiffness $κ > 0$ in J m $^{- 2}$ , and $H (q, p) = \sum_{i = 1}^{N} \frac{p_{i}^{2}}{2 m} + \frac{κ}{2} \sum_{i = 1}^{N} (q_{i + 1} - q_{i})^{2} .$ Adopt inverse temperature $β > 0$ in J $^{- 1}$ and fixed action reference $h_{0} = a_{0} > 0$ in J s under Declared example preparations and thermodynamic interpretation. Because the coordinates are resolved, use the reference $d q d p / h_{0}^{N}$ with no particle-exchange factorial. Define the partition integral $Z : = \int_{R^{N} \times R^{N}} \exp (- β H (q, p)) \frac{d q d p}{h_{0}^{N}},$ where $\exp$ is the real exponential of The real exponential function and the number $e$ by a power series. The unpinned position integral is infinite, so this model has no normalized canonical probability on the full position space.

Facts & Assumptions

Given: ${A C}_{ω}$ , the periodic-chain Hamiltonian, $m > 0$ kg, $κ > 0$ J m $^{- 2}$ , $β > 0$ J $^{- 1}$ , $h_{0} = a_{0} > 0$ J s, and the resolved-coordinate reference above.

[F1]

Declared example preparations and thermodynamic interpretation: when $Z$ is finite and positive, the canonical density is $Z^{- 1} \exp (- β H)$ relative to the chosen reference measure; the fixed action reference is part of the preparation, and resolved harmonic coordinates retain their declared labels without an added particle-exchange factorial.

[F2]

Under ${A C}_{ω}$ , Lebesgue measure of a box is its product volume, and a bounded Jordan measurable set has Lebesgue measure equal to its Jordan content (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included, Lebesgue outer measure is at most Jordan outer content, and a bounded Jordan measurable set is Lebesgue measurable with Lebesgue measure equal to its Jordan content).

[F3]

A $C^{1}$ diffeomorphism with nonzero Jacobian transforms Riemann integrals on compact Jordan sets by the absolute determinant (Change of variables for an injective $C^{1}$ map on a compact Jordan set).

[F4]

Every continuous real function on a nonempty compact metric space is bounded and attains a finite maximum (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

[F5]

The nonnegative Lebesgue integral is monotone under pointwise ordering (Monotonicity and nonnegative homogeneity of the nonnegative integral).

[A1]

The example adopts the displayed unpinned periodic-chain Hamiltonian and full $R^{N}$ displacement domain.

Verification

1.1givenA1F3algebra

Put $r_{1} : = 0$ and use coordinates $u : = q_{1}$ , $r_{i} : = q_{i} - q_{1}$ for $2 \leq i \leq N$ . The linear bijection $(u, r_{2}, \dots, r_{N}) \mapsto (q_{1}, \dots, q_{N}) = (u, u + r_{2}, \dots, u + r_{N})$ has determinant $1$ . With $r_{N + 1} : = r_{1}$ , every periodic difference satisfies $q_{i + 1} - q_{i} = r_{i + 1} - r_{i}$ , so the potential is independent of $u$ ; the momentum term is also independent of $u$ . On each compact coordinate box the change-of-variables supplier [F3] therefore gives the phase-volume Jacobian $1$ .

2.1F1F2F4F5step 1.1algebra∎

Under the declared reference convention [F1], choose any compact coordinate box $C$ of positive volume in $(r_{2}, \dots, r_{N}, p)$ . The Hamiltonian is continuous and hence bounded above on $C$ by a finite $H_{C}$ by [F4]. By [F2], $C$ has Lebesgue measure $vol (C)$ . On $u \in [- R, R]$ and $(r, p) \in C$ , the integrand is at least $e^{- β H_{C}}$ ; the determinant-one map in step 1.1 and [F2] show this slab contributes at least $2 R e^{- β H_{C}} vol (C) / h_{0}^{N}$ to the partition integral by [F5]. This lower bound is unbounded as $R \to \infty$ , so $Z = \infty$ and no normalization exists.

Remarks

This calculation identifies a translation zero mode. It makes no claim about separately confined or pinned models.

Depends on

Dependency tree · 2 levels

Sources

David Tong, Statistical Physics, §2.2.1, quadratic degrees of freedom