A higher Boltzmann distribution

Abstract

We characterize the classical Boltzmann distribution as the unique solution to a combinatorial Hodge theory problem in homological degree zero on a finite graph. By substituting for the graph a CW complex X and a choice of degree $d \le \dim X$, we define by direct analogy a higher dimensional Boltzmann distribution $\rho ^B$ as a certain real-valued cellular $(d-1)$-cycle. We then give an explicit formula for $\rho ^B$. We explain how these ideas relate to the Higher Kirchhoff Network Theorem of Catanzaro et al. (Homol Homotopy Appl 17:165–189, 2015). We also deduce an improved version of the Higher Matrix-Tree Theorems of Catanzaro et al. (Homol Homotopy Appl 17:165–189, 2015).

Appendices

Appendix 1: Kirchhoff–Boltzmann unification

For any fixed dimension d, the boundary operator $\partial: \, C_{d}(X;{\mathbb {R}}) \rightarrow C_{d-1}(X;{\mathbb {R}})$ factors as

$$\begin{aligned} C_{d}(X;{\mathbb {R}}) \,\xrightarrow{p}{} B_{d-1}(X;{\mathbb {R}}) \,\xrightarrow{i}{} C_{d-1}(X;{\mathbb {R}}) \end{aligned}$$

in which the first map is surjection and the second is an injection.

With respect to these inner product structures, one infers that the pseudo-inverse of $\partial\!\!:\, C_{d}(X;{\mathbb {R}}) \rightarrow C_{d-1}(X;{\mathbb {R}})$ is given by the composition

$$\begin{aligned} C_{d-1}(X;{\mathbb {R}}) \,\xrightarrow{{i_{E}^{ + } }}{} B_{d-1}(X;{\mathbb {R}}) \,\xrightarrow{{p_{W}^{ + } }}{} C_{d}(X;{\mathbb {R}}) \end{aligned}$$

consisting of the pseudo-inverses of i and p. (cf. Ben-Israel and Greville 2003, p. 48, ex. 17). Here, $p_W^+$ is the pseudo-inverse of p with respect to the modified inner product structure on $C_{d}(X;{\mathbb {R}})$ (cf. Remarks 3.2 and 3.5). Similarly, $i_E^+$ is the pseudo-inverse for i defined using the modified inner product on $C_{d-1}(X;{\mathbb {R}})$.

Therefore, all three pseudo-inverse formulas will follow from one, e.g., from the formula for the surjection, since the formula for an injection can be obtained by taking transposes and applying duality (Lemma 4.5), whereas the general pseudo-inverse is obtained by taking the composition.

For a tree $T \in {{\mathcal {F}}}_d(X)$, the composition

$$\begin{aligned} C_{d}(T;{\mathbb {R}}) \,{\xrightarrow{ \subset }{}} C_{d}(X;{\mathbb {R}}) \,\xrightarrow{\partial }{} B_{d-1}(X;{\mathbb {R}}) \end{aligned}$$

is an isomorphism. Let $\alpha _T$ denote its inverse. Let $\varphi _{T}: B_{d-1}(X; {\mathbb {R}}) \rightarrow C_{d}(X; {\mathbb {R}})$ be the composition

$$\begin{aligned} B_{d-1}(X; {\mathbb{R}}) \xrightarrow{\alpha _{T}}{{\cong }} C_{d}(T; {\mathbb{R}}) \xrightarrow{ \subset }{} C_{d}(X; {\mathbb{R}})\, . \end{aligned}$$

Then using Theorem 3.3 one has

$$\begin{aligned} p_W^+ = \tfrac{1}{\Delta _W}\sum _{T}w_T\varphi _{T}, \qquad \Delta _W := \sum _T w_T\, , \end{aligned}$$

(25)

where the sum is over spanning trees $T \in {{\mathcal {F}}}_d(X)$.

Similarly, the pseudo-inverse of the inclusion $i: B_{d-1}(X;{\mathbb {R}}) \rightarrow C_{d-1}(X;{\mathbb {R}})$ is obtained as follows: for a co-tree $L \in {{\mathcal {F}}}_{d-1}^{*}(X)$ the composition

$$\begin{aligned} B_{d-1}(X;{\mathbb {R}}) \rightarrow C_{d-1}(X;{\mathbb {R}}) \rightarrow C_{d-1}(X;{\mathbb {R}})/C_{d-1}(L;{\mathbb {R}}) \end{aligned}$$

is an isomorphism; let $\beta _L$ denote its inverse. Let $\zeta _{L}: C_{d-1}(X; {\mathbb {R}}) \rightarrow B_{d-1}(X; {\mathbb {R}})$ be the composition

$$\begin{aligned} C_{d-1}(X; {\mathbb {R}}) \rightarrow C_{d-1}(X;{\mathbb {R}})/C_{d-1}(L;{\mathbb {R}}) \xrightarrow {\beta_{L}} {{\cong}} B_{d-1}(X; {\mathbb {R}}) \end{aligned}$$

where the first map is the quotient homomorphism.

Then

$$\begin{aligned} i_{E}^{+} = \tfrac{1}{\nabla _E}\sum _{L}\tau _L\zeta _{L}, \qquad \nabla _E = \sum _L \tau _L\, , \end{aligned}$$

(26)

where the sum is over spanning co-trees $L \in {{\mathcal {F}}}_{d-1}^{*}(X)$.

Combining (25) and (26) with the pseudo-inverse composition property we arrive at a formula which encompasses both the Boltzmann splitting formula (Theorem A) and the higher Kirchhoff projection formula of Catanzaro et al. (2015, thm. A):

Theorem 5.3

(Kirchhoff–Boltzmann Projection Formula) The pseudo-inverse of the boundary operator $\partial\!\!:\, C_d(X;{\mathbb {R}}) \rightarrow C_{d-1}(X;{\mathbb {R}})$ with respect to the modified inner products defined by E and W is given by

$$\begin{aligned} \partial _{E, W}^{+} = \tfrac{1}{\Delta _W\nabla _E}\sum _{L,T}\tau _L w_T\sigma _{L, T}\, , \end{aligned}$$

(27)

where $\sigma _{L, T} = \varphi _{T}\circ \zeta _{L}: C_{d-1}(X; {\mathbb {R}}) \rightarrow C_{d}(X; {\mathbb {R}})$ and the sum is indexed over $L \in {{\mathcal {F}}}_{d-1}^{*}(X)$ and $T \in {{\mathcal {F}}}_d(X)$.

Remark 5.4

Theorem 5.3 gives a concrete expression for the average current in the case of periodic stochastic driving (in the adiabatic limit) (Chernyak et al. 2013). Let $\gamma$ be a smooth 1-dimensional cycle in the vector space of parameters (E, W) and let $\hat{x} \in Z_{d-1}(X;{\mathbb {Z}})$ be a $(d-1)$-cycle. Let $x = [\hat{x}] \in H_{d-1}(X;{\mathbb {Z}})$ be the associated homology class. Then the average current of $(\gamma ,[x])$ is defined to be the d-cycle

$$\begin{aligned} q := \int _{\gamma }A\, d\rho ^{\mathrm{B}} \in Z_d(X;{\mathbb {R}}) \, , \end{aligned}$$

where $A = A(E,W) = e^{-W}\partial ^{*}e^{E}{{\mathcal {L}}}_{E, W}^{-1}$ is the operator of equation (19) and $\rho ^{\mathrm{B}}$ is the higher Boltzmann distribution of x (see Chernyak et al. 2013) if $d = \dim X = 1$ and (Catanzaro 2016) for $d > 1$.). Then, after some straightforward algebraic manipulation using Theorem 5.3, we find

$$\begin{aligned} q = \sum _{L,T}\sigma _{L, T}(\hat{x})\int _{\alpha }d\varrho _{T}^{+} \wedge d\varrho _{L}^{-}\, , \end{aligned}$$

where $\varrho _{T}^{+} = \varrho _{T}^{+}(W) := w_T/\Delta _T, \varrho _{L}^{-} = \varrho _{L}^{-}(E) := \tau _L/\nabla _L$ and $\alpha$ is any smooth 2-dimensional chain in the space of parameters satisfying $\partial \alpha = \gamma$.

Appendix 2: connection with a result of Lyons

In this section we show how Corollary D is equivalent to the second formula of Lyons (2009, cor. 6.2). The latter result is the statement that the determinant of the restricted Laplacian ${\mathcal {L}}:\,B_{d-1}(X;{\mathbb {Q}}) \rightarrow B_{d-1}(X;{\mathbb {Q}})$ coincides with

$$\begin{aligned} \frac{h_{d-1}(X)h_{d-2}'(X)}{\theta _{d-2}(X)^2}\, , \end{aligned}$$

where

• $\theta _k(Y)$ is the order of the torsion subgroup of $H_k(Y;{{\mathbb {Z}}})$;

• $h_{d-1}(X) := \sum _T \theta _{d-1}(T)^2$ and the sum is indexed over all spanning trees T of X in dimension d;

• $h_{d-2}'(X) := \sum _L \theta _{d-2}(L)^2t_{d-1}'(L^c)^2\, ,$ where the latter sum is indexed over all spanning co-trees L of X in dimension $d-1$;

• the number $t_{d-1}'(L^c)$ that appears in the definition of $h_{d-2}'(X)$ is the order of the finite abelian group

  $$\begin{aligned} Z_{d-1}(X;{{\mathbb {Z}}})/(Z_{d-1}(X;{{\mathbb {Z}}})\cap B_{d-1}(X;{{\mathbb {Q}}}) + Z_{d-1}(L;{{\mathbb {Z}}})) \, . \end{aligned}$$

(Note: $\theta _k(Y)$ is the same as Lyons’ $t_k(Y)$.)

It is straightforward to check that the two results are equivalent if the identity

$$\begin{aligned} \frac{a_L}{\theta _{d-1}(X)}\,\, = \,\, \frac{\theta _{d-2}(L)t'_{d-1}(L^c)}{\theta _{d-2}(X)} \end{aligned}$$

(28)

holds for every such spanning co-tree L, where we recall that $a_L$ is the order of $H_{d-1}(X,L;{{\mathbb {Z}}})$.

The argument proceeds as follows. Unravelling the proof of Lyons (2009, lem. 6.1), one discovers that $t_{d-1}'(L^c)$ coincides with the order of the cokernel of the injective homomorphism

$$\begin{aligned} H_{d-1}(L;{{\mathbb {Z}}}) \rightarrow H_{d-1}(X;{{\mathbb {Z}}})/T_{d-1}(X;{{\mathbb {Z}}})\, , \end{aligned}$$

where $T_{d-1}(X;{{\mathbb {Z}}}) \subset H_{d-1}(X;{{\mathbb {Z}}})$ is the torsion subgroup.

Consider the commutative diagram

having exact rows and columns, where homology is taken with integer coefficients. The homomorphisms labelled by $\rightarrowtail$ are injective and those labelled by $\twoheadrightarrow$ are surjective. The injectivity of $H_{d-1}(L) \rightarrow H_{d-1}(X)$ and $H_{d-1}(L) \rightarrow H_{d-1}(X)/T_{d-1}(X)$ is implied by the spanning co-tree axioms.

Let $q_L$ denote the order of $H_{d-1}(X,L)/T_{d-1}(X)$. Truncating the third row in the middle, we obtain an exact sequence

$$\begin{aligned} 0\rightarrow \text {coker}(u) \rightarrow H_{d-1}(X,L)/T_{d-1}(X) \rightarrow \text {ker}(v) \rightarrow 0 \end{aligned}$$

which implies the identity

$$\begin{aligned} q_L = \frac{\theta _{d-2}(L)t'_{d-1}(L)}{\theta _{d-2}(X)}\, . \end{aligned}$$

(29)

Exactness of the middle column implies

$$\begin{aligned} a_L = q_L\theta _{d-1}(X)\, . \end{aligned}$$

(30)

The identity (28) now follows by inserting (30) into (29). $\square$