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Algebraic topology and the quantization of fluctuating currents

We give a new approach to the study of statistical mechanical systems: algebraic topology is used to investigate the statistical distributions of stochastic currents generated in graphs. In the adiabatic and low temperature limits we will demonstrate that quantization of current generation occurs.


Introduction
In statistical physics and chemistry, especially in the study of classical stochastic systems at the intermediate length scale, a master equation governs the time evolution of states, in which transitions between states are treated probabilistically. In its most compact form, the master equation isṗ = τ D H β p, where p(t) is a one parameter family of probability distributions on the state space, τ D is a constant that represents total driving time and H β is the master operator, which depends both on time t and a number β representing inverse temperature.
We will be interested in varying the parameters τ D and β. When τ D is made large, the duration of time it takes to traverse the driving path is large, and one refers to this process as adiabatic (or slow) driving. The limiting case τ D → ∞ is called the adiabatic driving limit. Similarly, one can consider the effects of low temperature on the system; the limiting case β → ∞ is referred to as the low temperature limit.
Associated with the formal solution of the master equation is an average current vector which represents the probability flux of a given initial distribution of states. In our first physics paper [CKS1], we argued that for generic periodic driving protocols, taking first the adiabatic limit and subsequently the low temperature limit results in an average current vector having integer components. This quantization phenomenon has been observed in a variety of applications, including electronic turnstiles, ratchets, molecular motors and heat pumps (cf. the bibliography of [CKS1]). One of the purposes of the current paper is to give this result a mathematically rigorous foundation. Our second aim is to explain how algebraic topology enters the picture in an essential way.
We now develop a mathematical formulation of our main results. Consider a particle taking a continuous time random walk on a connected finite graph Γ. The particle starts at a vertex i, say, and at a random waiting time it jumps to an adjacent vertex j where it waits again and so forth. Aside from the choice of inverse temperature β, such a process is determined by choosing a collection of real parameters, one assigned to each vertex (well energies) and to each edge (barrier energies) of the graph. The space of these parameters is denoted by M Γ ; it has the structure of a real vector space whose dimension d is the number of vertices plus the number of edges of Γ.
Current generation occurs when the parameters are allowed to vary in a one parameter family. 1 We consider such a family to be parametrized by an interval [0, τ D ], in which the number τ D represents total driving time. If the value of the parameters at the endpoints coincide, one obtains a periodic driving protocol; it can be represented as a pair (τ D , γ) in which γ : [0, 1] → M Γ is a smooth loop (equivalently, it is a smooth Moore loop). For each periodic driving protocol (τ D , γ) and each β we can associate a class Q τ D ,β (γ) ∈ H 1 (Γ; R) lying in the first homology of the graph with real coefficients. The class is defined in terms of the formal solution of the master equation and is called the average current generated by the triple (τ D , γ, β). Physically, the average current is a measurement of the "pumping" by external forces acting on the system.
The assignment γ → Q τ D ,β (γ) describes a smooth map where LM Γ is the space of smooth unbased loops in M Γ with the Whitney C ∞ topology. By taking the adiabatic limit τ D → ∞, and using the Adiabatic Theorem (Corollary 13.3), we obtain a smooth map Q β : LM Γ → H 1 (Γ; R) which does not depend on the parameter τ D . We call the latter the analytic current map.
If we subsequently take the low temperature limit β → ∞, it turns out that the resulting map is not everywhere defined.
Definition 1.1. A loop γ ∈ LM Γ is said to be intrinsically robust if there is an open neighborhood U of γ such that the low temperature limit Q := lim β→∞ Q β is well-defined and constant on U . The subspace of LM Γ consisting of the intrinsically robust loops is denoted byĽM Γ .
The main result of this paper is a quantization result for Q.
Theorem A (Pumping Quantization Theorem). The image of the map and a complex line bundle ξ over X (in terms of classifying spaces, this is the same thing as specifying a weak map X → BU (1)). When the weak equivalence h is understood, we sometimes drop it from the notation and simply refer to ξ as a weak complex line bundle over X.
Since h is a cohomology isomorphism, there is no loss in considering the first Chern class of ξ as lying in H 2 (X; Z). Now suppose that X = Y ×Z. Then slant product with c 1 (ξ) defines a homomorphism c 1 (ξ)/ : H 1 (Z; Z) → H 1 (Y ; Z) . Let S(Γ) = U (1) n be the n-torus, where n is the first Betti number of Γ.
Theorem F (Chern Class Description). There exists a weak complex line bundle ξ on the cartesian product S(Γ) ×M Γ such that

Preliminaries
Graphs. We fix a connected finite graph where Γ 0 is the set of vertices and Γ 1 is the set of edges. Here we are allowing multiple edges between vertices and also edges linking a vertex to itself (loop edges). The entire structure of Γ is then given by specifying a function d : Γ 1 → Γ (2) 0 which assigns to an edge the set of vertices which it connects (Γ (2) 0 denotes the two-fold symmetric product of the set of vertices). For convenience, we fix a total ordering for Γ 0 . Then d lifts to a map (d 0 , d 1 ) : Γ 1 → Γ 0 × Γ 0 in the sense that d(e) = {d 0 (e), d 1 (e)}, with d 0 (e) ≤ d 1 (e), where d 0 (e) = d 1 (e) if and only if e is a loop edge. The maps d i : Γ 1 → Γ 0 , for i = 0, 1 are called face operators.
The geometric realization of Γ is the one dimensional CW complex |Γ| given by the amalgamated union

Populations and currents.
Definition 2.1. The space of population vectors C 0 (Γ; R) is the real vector space with basis Γ 0 and the space of current vectors C 1 (Γ; R) is the real vector space with basis Γ 1 . If p is a population vector and i ∈ Γ 0 , then p i denotes the i-th component of p. Likewise, if J is a current vector α ∈ Γ 1 then J α denotes the α-th component of J.

The boundary operator
is given on basis elements by ∂(α) = d 0 (α) − d 1 (α). Then C * (Γ; R) is the cellular chain complex of Γ over the vector space of real numbers. The spaces C i (Γ; R) are smooth manifolds and ∂ is a smooth map which is a cellular chain analog of the divergence operator. If a current J lies in H 1 (Γ; R) := ker(∂), we say that it is conserved.
The subspaceC 0 (Γ; R) ⊂ C 0 (Γ; R) consisting of population vectors p such that i∈Γ 0 p i = 1 is called the space of normalized population vectors; these can be viewed as discrete probability density functions on the space of states Γ 0 . The subspaceC 0 (Γ; R) ⊂ C 0 (Γ; R) of those p such that i p i = 0 is called the space of zero population vectors.

Driving Protocols
Stochastic processes for periodic driving are governed by the "master equation" which is a certain linear first order differential equation acting on time dependent families of population vectors (see [vK, Ch. V]). Our master equation is a combinatorial analog of the Fokker-Planck equation in Langevin dynamics [vK, Chap. VIII].
The space of parameters. The space of parameters for Γ is the real vector space M Γ consisting of ordered pairs (E, W ) where E : Γ 0 → R and W : Γ 1 → R are real-valued functions. The function E is known as the set of well energies and W is is known as the set of barrier energies. We sometimes write (E i , W α ) for the value of (E, W ) at (i, α) ∈ Γ 0 × Γ 1 .
Remark 3.1. Notice that M Γ only depends on the number of vertices and edges of Γ, but not on the incidences. The subspace of "robust" parameters, which we introduce later, will depend in a crucial way on the incidence structure of the graph.
Periodic Driving. A driving protocol is a smooth path where the real number τ D > 0 plays the role of driving time. When γ(0) = γ(τ D ), we can view γ as a map C τ D → M Γ , where C τ D is the circle of length τ D . If in addition the latter map is smooth, we will say that γ is periodic. When τ D = 1, we say that γ is normalized.
Observe that a periodic driving protocol equivalent to specifying a pair (τ D , γ) ∈ R + × LM Γ in which γ is a normalized periodic driving protocol. Here LM Γ denotes the free (smooth) loop space of M Γ .
The master operator. Fix a real number β > 0. For a given (E, W ) ∈ M Γ we can form, for each i ∈ Γ 0 and α ∈ Γ 1 , the real numbers Letĝ : C 1 (Γ; R) → C 1 (Γ; R) be the linear transformation given by the diagonal matrix whose entries are g α . Similarly, letκ : C 0 (Γ; R) → C 0 (Γ; R) be given by the diagonal matrix with entries κ i .
In particular, for fixed β, we can view the master operator as defining a smooth map Remark 3.3. With respect to the inner product on C 0 (Γ; R) defined by u, v κ = uκv t , the master operator is self-adjoint. We infer that the eigenvalues of the master operator are real, and it is also easy to see that they are non-positive. When E = 0 = W , the master operator is just the graph Laplacian −∂∂ * . The master operator is also known as the Fokker-Planck operator to emphasize its natural interpretation as the discrete analogue of the Fokker-Planck operator in Langevin dynamics on smooth spaces.
where the convention is that H ij = 0 when S ij is empty, i.e., there is no edge connecting i and j. In particular, j∈Γ 0 H ij = 0 and H ij > 0 for i = j (compare [vK, p. 101]).
Remark 3.5. We offer comments on some distinctions in terminology between mathematics and physics. In the statistical mechanics literature, Γ is usually a simple graph (no multiple edges and no loop edges).
In this case the numbers H ij are called rates and describe a Markov process on Γ with transition matrix H (observe that H ij = k i,α with d(α) = {i, j} in this case). If X t denotes the state of the process at time t, then where the numerator appearing on the right denotes the conditional probability of transitioning to state j at time t + ∆t, given that one is in state i at time t. Because of Eq. (1), the rates satisfy the detailed balance equation which states that the net flow of probability from state i to state j is the same as that from state j to state i. This means that the Markov process is time reversible [vK, p. 109]. Conversely, if the process is time reversible, one can show that the parameters κ i and g α are, after possibly rescaling, in the form given by Eq.
(1). What we have described above is the notion of continuous time random walk on a graph. This is slightly more general than the notion of random walk considered in the mathematical literature (cf. [B, Chap. IX]). Mathematicians usually define a random walk to be a reversible Markov chain rather than the more general notion of reversible Markov process (the difference being that for Markov processes, one considers waiting times at the vertices as part of the walk).
The master equation. Fix a periodic driving protocol (τ D , γ), and β > 0. Then we have the associated one parameter family of master operators H(γ(t)) ∈ end R (C 0 (Γ; R)). The master equation is given bẏ The master equation governs the time evolution of probability: when p(t) is normalized, the component p i (t) represents the probability density of observing the state i at time t.
The Boltzmann distribution. Suppose V is a finite dimensional real vector space equipped with basis B. If E : B → R is a function, and β > 0 is a real number, we may form the normalized linear combination This is called the (normalized) Boltzmann distribution of the pair (E, β).
(In thermodynamics, β represents a multiple of inverse temperature: β = 1 k B T , where T is the temperature and k B is the Boltzmann constant.) The basis B identifies V with its dual space V * , so we are entitled to consider the function E as a vector lying in V having components E i . Then for fixed β, the Boltzmann distribution describes a smooth map is the open standard simplex with respect to the basis T (i.e., this map sends a vector E to its Boltzmann distribution). We say E is non-degenerate if there is a unique j ∈ T such that the j-th component E j of E is minimizing.
Lemma 3.6. Let f : [0, 1] → V be a smooth map with the property that f (t) is non-degenerate for every t ∈ [0, 1]. Then tends uniformly in t to the zero vector in the low temperature limit β → ∞.
Proof. As [0, 1] is compact, we only need to verify the statement pointwise, i.e., for each t ∈ [0, 1]. To avoid clutter we write E i := E i (f (t)). Then the i-th component of displayed derivative is Case (1): i is the minimizing vertex. In this instance, the denominator of Eq. (7) is the square of 1 + ( j =i e β(E i −E j ) ), where each E i − E j < 0.
Hence the denominator tends to 1 in the low temperature limit. As for the numerator of Eq. (7), when i = j, the term β(Ė j −Ė i )e β(E i −E j ) tends to 0 and when i = j it is 0. So the low temperature limit of Eq. (7) is 0.
Case (2): i isn't the minimizing vertex. In this instance at least one of E i − E j is positive and Eq. (7) is dominated by kβ/e cβ for a suitable choice of constants k and c with c > 0. The latter tends to zero in the low temperature limit by L'Hospital's rule.
The Boltzmann distribution for the population space. When B = Γ 0 , we have V = C 0 (Γ; R). The Boltzmann distribution in this case describes a smooth map whose value at (E, W ) depends only on E and β. It is not difficult to show that ρ B (E, W ) ∈ C 0 (Γ; R) is in the null space of the master operator H(β, E, W ) (compare [vK, p. 101]).

Current Generation
For a periodic driving protocol (τ D , γ) and β > 0, the instantaneous current at t ∈ [0, 1] is defined as where ρ(t) is the unique periodic solution of the master equation given by Proposition 13.1 below (here we are assuming that τ D is sufficiently large). Then the continuity equation is satisfied, in which J(t) plays the role of probability flux ( [vK, p. 193], [HJ]). The average current generated per period is This expression measures the net flow of probability in a single period [0, τ D ].
Average current in the adiabatic limit. In the adiabatic limit τ D → ∞, both J and Q can be expressed in terms of a certain differential C 1 (Γ; R)-valued 1-form A.
For each (E, W ) ∈ M Γ and β > 0, the negative of the restricted boundary map For fixed β and variable (E, W ), this defines a smooth map The proof of the following is immediate.
Lemma 4.1. The map A is uniquely characterized by the following properties: (1) The composition coincides with second factor projection, and (2) for all J ∈ H 1 (Γ), we have where −, − ĝ is the inner product on C 1 (Γ; R) defined by u, v ĝ = uĝv t .
Remark 4.2. The operator A defines the solution to Kirchoff's theorem on electrical circuits (see [B, p. 44]). Property (2) above amounts to Kirchoff's voltage law withĝ defining the resistance matrix.
An explicit formula for A is given as follows: choose a basepoint i ∈ Γ 0 . Given (W, E) ∈ M Γ and β > 0, define a linear transformation A e : C 0 (Γ; R) → C 1 (Γ; R) whose value at basis elements j ∈ Γ 0 is where the sum is over all spanning trees of Γ. The term Q T,j i is the element of C 1 (Γ; R) defined by the signed sum of edges along the unique path from i to j along T , where an edge has sign +1 if and only if its orientation coincides with the path. The term B T is the real number given by the T -component of the Boltzmann distribution whose vector space has basis the set of spanning trees of Γ, where the energy function is given by α∈T 1 W α . Then A e restricted to the subspaceC 0 (Γ; R) coincides with A.
Given a periodic driving protocol (τ D , γ) and β > 0, application of the normalized Boltzmann distribution gives a loop of normalized population vectors given by t → ρ B (γ(t)). Taking the time derivative, we obtain a loop of reduced population vectorṡ Then application of A to the pair (γ,ρ B (γ)) yields a loop of currents The following is then a straightforward consequence of the definitions combined with the Adiabatic Theorem 13.3. Proposition 4.3. Let β > 0 be fixed. Then in the adiabatic limit we have lim By appealing to Lemma 4.1, one sees that the image of the adiabatic limit of Q is contained in H 1 (Γ; R), so it defines a smooth map Remark 4.4. As our main results are stated in the adiabatic limit, there is no loss in pretending, even before taking the adiabatic limit, that the average current is given by the expression on the right-hand side of Proposition 4.3. With this change, the average current is defined without having to refer to either τ D or to solutions of the master equation.
Definition 4.5. The map given by Eq. (10) is called the analytical current map. Fig. 1. A graph with a total ordering of its edges and its associated σ-spanning tree.

Good parameters
Spanning trees. For each total ordering σ of the set of edges Γ 1 , we may define a spanning tree T σ for Γ by sequentially removing the links with the highest possible value in the ordering such that the remaining graph remains connected. Explicitly, let α 1 in Γ 1 be maximal. We discard α 1 if and only if the graph Γ \ α 1 : Repeating this process, the edges which are retained form a tree T σ .
Definition 5.1. The tree T σ given by the above procedure is called the spanning tree associated with σ, or simply the σ-spanning tree.
Example 5.2. Consider the graph with total ordering σ of its edges depicted at the top of Fig. 1. The associated σ-spanning tree is gotten as follows. If we remove the edge labeled 7, then the graph is connected, so we discard this edge. In Γ \ 7, the edge labeled 6 disconnects the graph when it is removed, so edge 6 is retained. Continuing in this fashion, the all edges but those labeled 3 and 7 are retained. This results in the spanning tree indicated by the path 1 → 2 → 4 → 5 → 6 indicated in the bottom of Fig. 1.
Example 5.3. A total ordering σ is determined by a choice of nondegenerate barrier energies W : For any such W and any spanning tree T we introduce the number When W is understood, we sometimes write w(T ) for w(T, W ).
Proposition 5.4. Let W : Γ 1 → R be nondegenerate. Let σ be the ordering of edges associated with W , as in Example 5.3. Then, for any spanning tree T ⊂ Γ with T = T σ , there is a spanning tree T ∈ Γ, so that w(T , W ) > w(T, W ).
Proof. Let (α 1 , . . . , α k ) and (β 1 , . . . , β k ) be the elements of Γ 1 \ (T σ ) 1 and Γ 1 \ T 1 , respectively, in the decreasing order with respect to σ, i.e., the barrier energies are decreasing from left to right. Let j be smallest index such that Consider the spanning subgraph Γ ⊂ Γ, obtained from Γ by withdrawing the edges α 1 , . . . , α j−1 , or equivalently β 1 , . . . , β j−1 . By the definition of T σ , the edge α j is not a bridge of Γ (i.e., its withdrawal does not disconnect the graph), and W β i < W α j for i ≥ j. Let T (1) and T (2) be the two trees obtained from T by withdrawing the edge α j . Then there is at least one edge, say β s , among β j , β j+1 , . . . , β k that connects T (1) to T (2) , since otherwise the edge α j would be a bridge of Γ. Therefore, by replacing the edge β s with α j in T results in another spanning tree, denoted T that obviously satisfies the condition of the proposition, since W α j > W βs .
Proposition 5.4 gives an immediate characterization of σ-spanning trees in terms of the function w(−, W ). Let T Γ denote the set of spanning trees of Γ.
Corollary 5.5. With σ and W as in Proposition 5.4, the σ-spanning tree T σ is the unique maximizer of the function w(−, W ) : T Γ → R.
Remark 5.6. One can restate the last corollary so as to depend only on σ: For α ∈ Γ 1 set W σ (α) = k if α is the k-th element in the partial ordering given by σ. Now define ω(T, σ) := w(T, W σ ). Then by Corollary 5.5, T σ is the unique maximizer of ω(−, σ) : T Γ → R.
Remark 5.7. There is a useful alternative characterizing property of the σ-spanning tree T σ associated with a nondegenerate barrier function W : for each withdrawn edge (i.e., an edge not in T σ ) with d(α) = {i, j}, the barrier W α is higher than any of the barriers associated with the edges of the unique embedded path which connects i to j inside T σ . For this reason we named T σ the minimal spanning tree of W in [CKS1]. (1) there is only one absolute minimum for E : Γ 0 → R, or (2) the function W : Γ 1 → R is one-to-one, i.e., the edges are distinguished by their barrier energies. (In this instance we say that W is nondegenerate.) We callM Γ the space of good parameters.
Let U be the set of (E, W ) satisfying the first condition and let V be the set of (E, W ) satisfying the second. Then where U and V are open. Each connected component of U is defined by specifying a vertex v ∈ Γ 0 , whereas each connected component of V is given by specifying a total ordering σ of Γ 1 . Consequently, we have decompositions into connected components For each vertex v ∈ Γ 0 , let B 1/3 (v) be the set of points in |Γ| which have distance < 1/3 from v in the natural metric on |Γ| that gives every edge a length of 1. Let N v ⊂M Γ × |Γ| be the subspace given by . Then the second factor projection Then the projection N U → U is also a homotopy equivalence. Call this projection p U . For a given σ, we let N σ ⊂M Γ × |Γ| be the subset consisting of V σ ×|T σ |(1/3), where T σ is the σ-spanning tree and |T σ |(1/3) consists of the points of |Γ| whose distance to |T σ | is < 1/3. Then the projection N σ → V σ is a homotopy equivalence (it is the cartesian product of V σ with a metric tree). Set then a straightforward application of the gluing lemma [tD] shows that the first factor projection is a homotopy equivalence.
Definition 5.8. For good parameters, the weak mapq is given by where p 2 denotes the second factor projection.

A weak form of the Pumping Quantization Theorem
Recall the decompositionM where j ranges through the elements of Γ 0 and σ ranges through the set of total orderings of Γ 1 . Given a loop γ ∈ LM Γ , it will be convenient in what follows to think of γ as a smooth map C →M Γ , where C denotes the circle of radius 1/(2π). Let I ⊂ C be a closed arc. The contribution along I to the analytical current map is then given by the integral where we parametrize I with respect to arc length. That is, if I 1 , . . . I k is a simplicial decomposition of C into closed arcs, then Assume that γ(I) ⊂ U ; in this instance we say C is of type U .
Lemma 6.1. If I is of type U , then in the low temperature limit the contribution along I to Q β (γ) is trivial.
Proof. On I the function E : Γ 0 → R has a unique absolute minimum v. As β tends to ∞, the value of Boltzmann distribution ρ B restricted to arc I tends to v. This is because the component of ∈ Γ 0 in the Boltzmann distribution is e −βE i e −βE i and the latter tends to zero on if = v and one if = v when E v is the unique minimum. Consequently, as β tends to ∞, the time derivativė ρ B tends to zero on I (by Lemma 3.6).
The contribution to the current of γ along I is given by the integral (using Proposition 4.3). When β tends to infinity, this expression tends to zero.

Now consider a closed arc
Lemma 6.2. If I is of type V , then in the low temperature limit the contribution along I to Q β (γ) is an element of C 1 (Γ; Z).
T is the T -component of the Boltzmann distribution for the vector space whose basis is the set of spanning trees of Γ. Recall also that the instantaneous current J(t) is defined as A(ρ B ), where ρ B in this case denotes the Boltzmann distribution for C 0 (Γ; R). Hence, insertingρ B into the expression for A e gives (12) Recall that the average current is given by 1 0 J(t). In particular, the contribution along I is given by d c J(t)dt (where we are parametrizing I with respect to arc length and the limits of integration come from the parametrization). Hence, integrating both sides of the last display, we obtain Since Q T,j i is an integer valued 1-chain, it will suffice to prove that tends to an integer as β tends to ∞. Using integration by parts, we may rewrite this expression as By Lemma 3.6,ρ B T tends to zero as β → ∞. Hence the constribution to the current along I in the low temperature limit is determined by Since γ(∂I) ⊂ U , we deduce by the argument of Lemma 6.1 that the low temperature limit of B T ρ B T | b a is an integer. As an immediate corollary, we obtain a weak version of Theorem A: Theorem 6.3 (Weak Quantization). If γ ∈ LM Γ , then the low temperature limit lim β→∞ Q β (γ) is defined and lies in the integral lattice

The Discriminant Theorem and Robust Parameters
SetD := M Γ \M Γ . We will first show that the one-point compactificationD + has the structure of a regular CW complex. By definition,D is the subspace of M Γ consisting of pairs (E, W ) such that E : Γ 0 → R has more than one absolute minimum and W : Γ 1 → R is not one-to-one.
Height functions arise in the following situation.
Example 7.2. Given (E, W ) ∈D, we write h 0 (i) = 1 if and only if i is a minimum for E, and otherwise we set h 0 (i) = 2. We define h 1 : Γ 1 → {1, . . . , n} to be the unique surjective function characterized by to be the set of all (E, W ) ∈D whose associated height function is h, as in Example 7.2. ThenD = h C(h) as sets. Note that C(h) is non-empty if and only if h −1 0 (1) has more than one element and h −1 1 (k) has more than one element for some k. Let D(h) denote the closure of C(h) inD. Proposition 7.3. Assume that C(h) is non-empty. Then the onepoint compactification D(h) + is homeomorphic to a disk of dimension m + n, where m = 1 + |Γ 0 \ h −1 0 (1)| and n is as above. Proof. The space C(h) coincides with the cartesian product C 0 (h) × C 1 (h), where C 0 (h) consists of those E : Γ 0 → R associated with h 0 and C(h 1 ) consists of W : Γ 0 → R associated with h 1 . Likewise D(h) coincides with the cartesian product D(h 0 ) × D(h 1 ).
Note that D(h 0 ) is canonically homeomorphic to the space P m consisting of m-tuples or real numbers (x 1 , x 2 , . . . , x m ) such that x 1 ≤ x k for all k > 1. The homeomorphism is given by mapping E to the map Γ 0 /∼ → R, where where ∼ denotes the equivalence relation on Γ 0 defined by i ∼ j if and only of h 0 (i) = 1 = h 0 (j). As Γ 0 /∼ has cardinality one more than the number of non-minima of E, such functions are identified with m-tuples of real numbers. The operation (z 1 , . . . , z m ) → (z 1 , z 2 + z 1 , . . . z m + z 1 ) defines a homeomorphism to P m from the space of of m-tuples (z 1 , . . . , z m ) such that z 1 ∈ R and z k ≥ 0 for all k > 1. It is easy see that the one-point compactification of the latter space is homeomorphic to D m . Hence, D(h 0 ) + is homeomorphic to D m .
Similarly, the space D(h 1 ) is canonically homeomorphic the space Q n consisting of n-tuples or real numbers (x 1 , x 2 , . . . , x n ) such that x 1 ≤ x 2 ≤ . . . x n . The operation (z 1 , . . . , z n ) → (z 1 , z 1 + z 2 , . . . , n i=1 z i ) defines a homeomorphism to Q n from the space of n-tuples (z 1 , . . . , z n ) for which z 1 ∈ R and z i ≥ 0 for i > 1. The one-point compactification of the latter space is identified with D n . Consequently, D(h 1 ) + is homeomorphic to D n . Finally, A top dimensional cell C(h) ofD + is given by a height function h in which • h −1 0 (1) has precisely two elements; • there is a k ≤ n such that h −1 1 (k) has precisely two elements, and if i = k we have h −1 1 (i) is a singleton for 1 ≤ i ≤ n. Robust parameters. Alexander duality applied to the inclusionD Let C d−2 (D + ) be the cellular cochain complex ofD + over the integers in degree d − 2. This is the free abelian group with basis given by the set of (d − 2)-cells ofD + . We consider the composition where q :M Γ → |Γ| is the weak map defined in Eq. (11) above. This homomorphism is naturally identified with a H 1 (Γ; R)-valued chain and it is trivial to check that φ is a cycle. For a ∈ C d−2 (D + ), let a, φ ∈ H 1 (Γ; R) denote the effect of applying the homomorphism (14) to a.
Definition 7.6. A (d − 2)-cell C(h) ofD + is said to be essential if C(h), φ ∈ H 1 (Γ; R) is non-trivial, where we consider C(h) as an element of C d−2 (D + ). A cell C(h) ofD + of any dimension is inessential if it is not contained in the closure of an essential (d − 2)-cell.
DefineĎ to be the closure of the union of the essential (d − 2)-cells ofD.  lower dimensional cells, each of these having boundary a union of lower dimensional cells and so on. In particular,D + is a union of interiors of certain cells, and this union is closed. Hence it is a subcomplex.
Definition 7.8. SetM Then we have an inclusionM Γ ⊂M Γ . We callM Γ the space of robust parameters.
Example 7.9. Let Γ denote the graph displayed in Figure 2. In this case, the vector space of parameters M Γ is identified with R 4 and the discriminantD is identified with the diagonal inclusion R 2 ⊂ R 4 given by (x, y) → (x, x, y, y). Taking the one-point compactification of this inclusion, we obtain an inclusion S 2 ⊂ S 4 that is identified with D + ⊂ M + Γ . The complement of this inclusion has the homotopy type of S 1 . Consequently, there is a homotopy equivalencȇ In fact, one can make this identification precise using the loop γ given by the length one periodic driving protocol γ(t) = (cos 2πt, 0, sin 2πt, 0) (one verifies this by showing that linking number of γ with S 2 ⊂ S 4 is ±1). Consequently, the first homology group ofM Γ is generated by the homology class [γ]. The effect ofq * : H 1 (M Γ ) → H 1 (Γ) on this loop is to produce a homology generator of H 1 (Γ; R) ∼ = Z (see [CKS1] for details), so the weak mapq :M Γ → |Γ| is a weak homotopy equivalence. In particular, the unique 2-cell ofD + ∼ = S 2 is essential, and we infer in this instanceM Γ =M Γ .
Remark 7.10. With little difficulty, Example 7.9 can be generalized to show thatq * : H 1 (M Γ ) → H 1 (Γ) is non-trivial whenever Γ has nontrivial first Betti number. We omit the details.
The weak mapq. Let C(h) denote a (d − 2)-cell ofD + , and let x ∈ C(h) be its center. We choose a small closed 2-disk meeting C(h) normally at x. The boundary of this disk represents a periodic driving protocol γ in the space of good parameters. As an illustration, we have depicted such a disk in Figure 3, associated with the graph Γ = >− . If the disk is sufficiently small then it is partitioned into four regions, the interior of each either contained in U or V , which is shown in Figure  3 as green and red respectively. 2 Each green sector corresponds to a vertex i or j of Γ, each giving a minimum for {E k } k∈Γ 0 . Likewise, each red sector corresponds to a preferred maximal spanning tree, and these are indicated next to each such sector together with the location of the vertices i, j. The topological current associated with the driving protocol γ is defined by the cycle obtained by joining together two paths connecting i with j (cf. §8). Each path is defined by moving along one of the spanning trees starting at i and terminating at j (for example, in Figure 3 one obtains the cycle , which is nontrivial). In particular, the cell C(h) is inessential if and only if this cycle is trivial. However, even more is true: since each path used to form the cycle is embedded, it is straightforward to check that the loop in Γ given by gluing these two paths together is null homotopic if and only if the two paths coincide. Consequently, if σ is inessential then the weak mapq extends to the space given by attaching a two cell toM Γ along γ. If we repeat this construction for every inessential (d − 2)-cell, we obtain a space which is homotopy equivalent to the space of robust parametersM Γ . We have therefore shown that the weak mapq admits an extension to the space of robust parameters. We denote this extension byq.
However, in order to prove the Pumping Quantization Theorem, it will more convenient to give a concrete extension of the weak mapq to all ofM Γ , rather than just a model for the extension up to homotopy. This construction will be described in the next section.
Example 7.11. In the graph depicted by Figure 4, the edge containing vertices i and j is in every maximal spanning tree. Therefore, the discriminantD for this graph has an inessential 4-cell.
A barrier resolution r h of h enables one to associate a total ordering σ r of the set Γ 1 . We saw in §5 how to obtain a spanning tree T σr for Γ associated with σ r . Let where the intersection is indexed over the set of barrier resolutions r of h. Then F h is a forest, i.e., a (possibly empty) disjoint union of trees.
Proposition 7.13. A cell C(h) is inessential if and only if all elements of h −1 0 (1) belong to the same connected component of F h . Proof. Consider a cell C(h) that satisfies the condition that all i ∈ h −1 0 (1) belong to the same connected component T h of the forest associated with C(h). Then T h is a tree. Consider an arbitrary top dimensional cell C(ĥ), so that C(h) ⊂ D(ĥ). The cell C(ĥ) is uniquely identified by two distinct vertices i, j ∈ h −1 0 (1), and two distinct edges α and β with h 1 (α) = h 1 (β), together with a height function h 1 of height |Γ 1 | − 1 withĥ 1 (α) =ĥ 1 (β), i.e., W α = W β is the only edge degeneracy in C(ĥ). Then we haveĥ 0 (k) = 1 for k = i, j andĥ 0 (k) = 2, otherwise. Consider a point (E, W ) ∈ C(ĥ), and let γĥ be a small loop that goes around (E, W ) without intersecting C(ĥ), i.e., staying within the good parameter spaceM Γ . Using the notation of Eq. (9), we have for the topological current whereh α andh β are the two possible barrier resolutions ofĥ with h α >h β andh α <h β , respectively. Thenh α andh β are also barrier resolutions of h, and, therefore, T h ⊂ Th α and T h ⊂ Th β , which implies Q . Therefore, Q(γĥ) = 0, due to Eq. (15), so that C(h) is inessential.
To prove the converse, let C(h) be any inessential cell. For an arbitrary pair of edges α and β letĥ 1 : Γ 1 → {1, . . . , |Γ 1 | − 1} be any surjection such thatĥ 1 (α) =ĥ 1 (β). Then there are two possible barrier resolutionsh α andh β ofĥ, as described above. For arbitrary distinct vertices i and j letĥ 0 : Γ 0 → {1, 2} be the function defined bŷ Thenĥ = (ĥ 0 ,ĥ 1 ) is a height function, and C(ĥ) is top dimensional. If C(ĥ) is inessential then Q(γĥ) = 0 and Eq. (15) implies that the minimal paths that connect i to j along the spanning trees Th α and Th β are identical. Furthermore, if D(ĥ) ⊃ C(h), then C(ĥ) is inessential. Consequently, the set consisting of the minimal paths that connect i to j inside the spanning trees Th associated with all barrier resolutions h of h consists of a single element. Denote this unique path by l h,ij . Then for any two distinct vertices i, j ∈ h −1 0 (1) the path l h,ij belongs to all spanning trees Th, and therefore the forest associated with C(h). We infer that all vertices that belong to h −1 0 (1) belong to the same connected component of the forest.

The weak mapq
Proposition 7.13 shows that to any inessential cell C(h) one can assign, in a preferred way, a tree that contains h −1 0 (1). This tree will be denoted T C(h) and referred to as the tree associated with the inessential cell C(h). We now cover each open cell C(h) with an open set Y h ⊃ C(h) that consists of all (E, W ) ∈ M Γ that is characterized by the property that if h 0 (i) < h 0 (j) and h 1 (α) < h 1 (β), then E i < E j and W α < W β , respectively, for all i, j ∈ Γ 0 and α, β ∈ Γ 1 . Setting Y = h Y h , where the union goes over all inessential cells we obtain an open coverM of the space of robust parameters. (1/3) is the open regular neighborhood of |T C(h) | given by adjoining half open subintervals of length 1/3 that correspond to those edges not in T C(h) but which contain a vertex of it.

For a given height function
Then the projection N h → Y h is a homotopy equivalence. We further define N Y = h N h , and if we seť whose arrows are given by the first and second factor projections respectively. A straightforward application of the gluing lemma which we omit shows that the projection map p 1 is a homotopy equivalence. We infer that the Eq. (18) describes a weak map which we sometimes write asq By construction, the restriction ofq toM Γ coincides withq.
For the sake of completeness we now sketch a proof thatq coincides up to homotopy with the extensionq ofq that was described in the previous section.
Lemma 8.1. The homotopy class of the weak mapq coincides with the homotopy class of the weak mapq given by gluing in 2-cells.
Proof. The result will follow from the existence of a commutative dia-gramM in which the vertical maps are homotopy equivalences and the left horizontal maps (denoted p 1 in each case) are also homotopy equivalences. The bottom maps labelled p 1 and p 2 comprise the weak mapq. The top maps labelled p 1 and p 2 define the extensionq. The spaceM Γ is obtained by attaching suitable two cells toM Γ ; it comes equipped with a decompositionM Γ = U ∪ V ∪Ŷ whereŶ consists of the set of inessential 2-cells, each which is labelled asD h , where h ranges over height functions that define a top dimensional inessential cell. Similarly, we defineN ⊂M Γ × |Γ| to be the union of N h =Ŷ h × |T C(h) |(1/3). Then the projections onto each factor explicitly define the extension ofq described in §7.
The vertical maps in the diagram are given as follows. The homotopy equivalenceM Γ →M Γ is given by the identity onM Γ and by mapping each closed 2-cellĎ h that is attached toM Γ homeomorphically to a small closed 2-cellĎ h that intersects C(h) transversely at its center (the boundary ofĎ h is prescribed to having linking number +1 with C(h)). A similar argument which we omit defines the homotopy equivalenceN →Ň .
Remark 8.2. For the proof of the Pumping Quantization Theorem, it will be convenient to put the open sets U j , V σ , and Y h on equal notational footing. This can be done by changing the notation to where h = (h 0 , h 1 ) is such that h 0 (k) = 2 for all k ∈ Γ 0 , and h 1 : Γ 1 → {1, . . . , |Γ 1 |} is given by h 1 is defined in the obvious sense by the total ordering σ. Similarly, we set where in this instance h = (h 0 , h 1 ) is given by h 0 (k) = 1 for k = j and h 0 (k) = 2 otherwise, and h 1 is the function with constant value 1. Note that these notational changes necessitate a more flexible notion of height function, which we will call an extended height function.
Note that the tree T C(h) that corresponds to V σ is just the σ-spanning tree T σ , whereas T C(h) that corresponds to U j consists of the single vertex j and no edges. With the above extension Eq. (17) readsŇ = N Y with N Y = hŇ h , where the union indexed over the set of extended height functions.

The Representability Theorem
Proof of Theorem B. It is clear that LM Γ ⊂ĽM Γ , so it suffices to prove the reverse inclusion. The proof will be by contradiction. Let γ ∈ĽM Γ . The idea is to modify γ along a small arc in such a way that the value of the current changes.
Suppose there is an s ∈ [0, 1] such that y = γ(s) ∈Ď. Then y is in the closure of an essential cell. Let > 0 be small. Choose a point x in the interior of this cell such that |y − x| ≤ with respect to a choice of norm on M Γ . Let V be an open neighborhood of γ in LM Γ on which Q is well-defined and constant.
If is sufficiently small, we can construct a smooth loop ω ∈ V which coincides with γ off of (s − , s + ), and inside this neighborhood ω winds once around a small disk D meeting the essential cell transversely at x in such a way that D \ x ⊂M Γ and Q(∂D) is nontrivial. Then topologically, ω is a loop obtained by concatenating γ with ∂D. As the current Q is additive, we find that Q(ω) = Q(γ) + Q(∂D). Consequently, Q(ω) = Q(γ). This contradicts the assumption that Q is constant on V .

The Pumping Quantization and Realization Theorems
Consider a closed arc I = [a, b] ⊂ C of our unit length circle C such that γ(I) ⊂ Y h for some h. Obviously, for c = a, b there is a welldefined low-temperature limit ρ (c) = lim β→∞ ρ B (γ(c); β), represented by a normalized constant function on its support supp(ρ (c) ) ⊂ h −1 0 (1). We further simplify the notation by using T h = T C(h) and chose some arbitrary base vertex i(h) in the tree T h for each relevant height function h.
Lemma 10.1. The contribution along I to the current Q β (γ) in the low-temperature limit is given by Proof. Using the explicit expression for the current based on the Kirchhoff theorem given Eq. (12) one can show where K h = {(j, T ) : h 0 (j) = 2 or T h T }. To derive Eq. (21) we first apply integration by parts to the explicit expression for the current, followed by representing the sum over the graph vertices j and spanning trees T as a sum over (j, T ) with h 0 (j) = 1 and T ⊃ T h , and the remaining terms. We also make use of the fact that provided j ∈ h −1 0 (1), which implies j ∈ (T h ) 0 , and T h ⊂ T , we have Q T,j i(h) = Q T h ,j i(h) , i.e., the contribution to the current does not depend on the spanning tree T . Then Eq. (20) follows from the following properties that hold inside Y h , and are verified directly. For j ∈ h −1 0 (2) we have lim Proof of Theorems A and E. Let I 1 , . . . , I k be a simplicial decomposition of S 1 into closed arcs, with 1, . . . , k ∈ Z/k, and I m = [a m−1 , a m ], so that γ(I m ) ⊂ Y h m for some set h 1 , . . . h k of (extended) height functions. Applying Lemma 10.1, and more specifically Eq. (20), followed by re-grouping the terms in the sum over the arcs we obtain The expression in the parenthesis on the right side of Eq. (25) does not depend on j. Indeed, this assertion needs only to be checked for for another vertex j which lies in T hm ∩ T h m+1 . In this instance the unique path running from j to j which is contained in T hm ∩ T h m+1 determines a one-chain c such that Q T h m ,j i(h m ) = Q T h m ,j i(h m ) + c and likewise Q If we also account for the normalization condition for ρ (am) , we can replace summation over j by choosing any vertex j m ∈ (h m 0 ) −1 (1) ∩ (h m+1 0 ) −1 (1) and then recast Eq. (25) in the form The right side of Eq. (26) is clearly an integer valued one-chain, so the proof of Theorem A is complete.
We now turn to the proof of Theorem E. With respect to the above situation, consider the free loop : S 1 → |Γ| defined as follows: when the parameter s ∈ S 1 changes from the center of the arc I m to its end a m , (s) goes from i(h m ) to j m along the unique minimal length path in the tree T h m . When s changes from a m to the center of the arc I m+1 , (s) goes from j m to i(h m+1 ) along the unique minimal length path in the tree T h m+1 . It is easy to see that the right side of Eq. (26), considered as an element of H 1 (|Γ|), is the image of under the map L|Γ| → H 1 (|Γ|) that associates with a free loop its corresponding homology class. On the other hand it is also easy to see that (γ, ) : S 1 → (M Γ × |Γ|) has image inŇ ⊂M Γ ×|Γ|, and we infer (γ, ) ∈ LŇ . By a straightforward inspection of the definitions we see that the right side of Eq. (26) is given byQ(γ) (as defined above in Definition 10.2), which completes the proof.
A combinatorially defined line bundle. We refer the reader to discussion of §8, especially Remark 8.2. Recall thať is a covering by open sets where h = (h 0 , h 1 ) ranges over extended height functions. Associated with Y h one has a tree T h := T C(h) such that h −1 0 (1) ⊂ T C(h) . Fix a basepoint vertex i for T h (cf. Lemma 10.1) For (E, W ) ∈ Y h and λ ∈ C 1 (Γ; U (1)), we associate a complex line in C n , where n is the cardinality of Γ 1 . For any vertex j of the tree T h , we have a minimal path P T h ,j i from i to j which is contained in T ; this path defines the integer value 1-chain Q T h ,j i (cf. Remark 4.2). Let C[Γ 0 ] denote the complex vector space with basis Γ 0 . Then we obtain a non-zero vector where s(α) = ±1 according as to whether the direction of the path coincides with the orientation of α (this sign coincides with the coefficient appearing of α in Q T h ,j i ). Let Cv ⊂ C[Γ 0 ] denote the complex line spanned by this vector.
Lemma 11.2. If we choose a different basepoint vertex the complex line Cv remains unchanged.
Proof. The amalgamation of the minimal length paths P T h ,i i and P T h ,j i produces a new path P T h ,i i P T h ,j i from i to j. If this path is minimal, then it is P T h ,j i and clearly, we have If the amalgamated path isn't minimal, then this formula still holds because the factors corresponding to indices occurring more than once cancel. This gives independence with respect to the basepoint vertex, as the first factor on the right is independent of j.
For fixed h the assignment (E, W, λ) → Cv(h, E, W, λ) describes a line bundleξ h over C 1 (Γ; U (1)) × Y h . In fact, it is straightforward to check thatξ h is trivializable. We now use the clutching construction to glue these line bundles together as h varies. This will produce a line bundle overξ over C 1 (Γ; U (1)) ×M Γ . To check this, it suffices to establish the following.
Lemma 11.3. Given height functions h and h , let denote the inclusions. Then there is an isomorphism of line bundles Furthermore, this isomorphism satisfies the cocycle condition φ ac = φ ab φ bc .
Proof. Associated with (λ, E, W ) ∈ C 1 (Γ; U (1)) × Y h and a basepoint vertex i for T h ∩ T h , we have a non-zero vector v which is defined by Eq. (27). To indicate the dependence of this vector on h, let us The cocycle condition is then immediate.
In order to prove Theorem 11.4 we digress to explain how holonomy relates to the homomorphism given by slant product with the first Chern class. If ξ is a complex line bundle over a connected space B and structure group U (1), we have the holonomy map (29) h ξ : LB → U (1) .
In fact, h ξ can be chosen in such a way that if we choose a basepoint of B and restrict h ξ to the based loop space ΩB, we can deloop to a map B → BU (1) = CP ∞ that classifies the bundle ξ and hence the Chern class c 1 (ξ). Consequently, if we choose h ξ in this way, it determines the Chern class. Now suppose B = X × Y . Then we can restrict h ξ to the subspace X × LY ⊂ LX × LY = L(X × Y ) and take the adjoint to obtain a map where F (X, U (1)) is the function space of maps from X to U (1). Then the diagram commutes, where the left vertical map sends a loop to its homology class and the right vertical map sends a function to its homotopy class considered as an element of H 1 (X; Z) = [X, U (1)].
Proof of Theorem 11.4. By Eq. (28) it suffices to show that c 1 (ξ)/ coincides with the homomorphism Suppose we are given (λ, γ) ∈ C 1 (Γ; U (1))×LM Γ . As done previously, we partition S 1 into closed arcs [a k , a k+1 ] for 0 ≤ k ≤ n with a n+1 ≡ a 0 , such that the projection of such an arc intoM Γ is contained in a neighborhood of type Y h k−1 . Choose a basepoint vertex i k lying in the intersection T k ∩ T k+1 , where T k denotes the tree associated with Y h k . Then we have a minimal length path P T k ,i k+1 i k from i k to i k+1 and the product describes a map C 1 (Γ; U (1)) → U (1) that gives the holonomy around γ (here we are using Lemmas 11.2 and 11.3).
In the special instance of λ ∈ C 1 (Γ; U (1)) which is identically 1 except for a single edge α, the value of the map C 1 (Γ; U (1)) → U (1) at λ is given by λ q α where q represents the net number of times α is traversed, with orientation taken into account. Consequently, if we identify H 1 (C 1 (Γ; U (1)); Z) with C 1 (Γ; Z), then λ q α is identified with the chain qα. It follows that the map C 1 (Γ; U (1)) → U (1) defined by Eq. (31) corresponds to the integer cycle in C 1 (Γ; Z) given by On the other hand, the paths P T k ,i k+1 i k describe a lift of γ : S 1 →M Γ through the spaceŇ appearing Eq. (17) (roughly, one defines the lift by mapping the midpoint a k of the arc [a k , a k+1 ] to the point (γ(a k ), i k ) and uses P T k ,i k+1 i k to connect these points). Then application of the projection map p 2 :Ň → |Γ| to the given lift produces a map S 1 → |Γ| that representsq * ([γ]) ∈ H 1 (Γ; Z) ⊂ C 1 (Γ; Z). From this description it is straightforward to check thatq * ([γ]) coincides with the element defined by Eq. (32) (cf. Eq. (9)).

The ground state bundle: a conjecture
By coupling the master operator with elements of the torus H 1 (Γ; U (1)) one can extend the master operator to a self-adjoint operator over the complex numbers. This extension is called the twisted master operator; its eigenvalues are real and non-positive. The eigenspace associated with the maximum non-zero eigenvalue is called the ground state. One may use the twisted master operator to define another weak complex line bundle η, this time over H 1 (Γ; U (1)) ×M Γ , whereM Γ ⊂ M Γ is characterized by the condition that the ground state at each point is non-degenerate, meaning that it has rank one. Then roughly, η is defined by taking the ground state at each point of the base. We call this the ground state bundle. Arguments from physics suggest that the ground state bundle is equivalent to the weak complex line bundle ξ that was defined in the previous section. In what follows we will formulate this idea as a pair of conjectures.
The twisted master operator. The twisted master operator, defined below, is a smooth map where C 0 (X; C) is the complex vector space with basis Γ 0 . It extends the master operator in the sense that where 1 ∈ C 1 (Γ; U (1)) is the function with constant value 1 ∈ U (1) and where we are interpreting the right side of this identity using extension of scalars.
Let the gauge group G Γ act on end C (C 0 (X; C)) via conjugation and trivially on M Γ . The following is then a formal consequence of the definitions.
Lemma 12.1 (Gauge Symmetry). The twisted master operator is G Γequivariant, i.e., for h ∈ G Γ , we havē In particular, for each (E, W ) ∈ M Γ , the spectrum of H(β, E, W ) = H(1, β, E, W ) is invariant with respect to the action of the gauge group.
Definition 12.2. Let A : V → V be a self-adjoint linear transformation of a finite dimensional complex vector space V , all of whose eigenvalues are non-positive. Then the ground state L is the eigenspace for maximal eigenvalue of A. We say that A has nondegenerate its ground state L has rank one.
An analytically defined weak line bundle. Define an open subset to be the set of those (β 0 , E, W ) such that for every λ ∈ C 1 (Γ; U (1)) and every β ≥ β 0 , the twisted master operator has a non-degenerate ground state.
Proof. Let V = C 0 (Γ; C) and let L 1 (V, V ) denote the space consisting of complex linear self-maps of V having corank one. Then L 1 (V, V ) is a smooth manifold of real dimension 2|Γ 0 | 2 − 2 (see [GG,prop. 5.3]). The operation which sends a complex linear self-map to its null space defines a smooth map whose target is the projective space of complex lines in V . The composition is therefore smooth and the pullback of the tautological line bundle over P 1 (V ) gives the projection p.
Letη be the complex line bundle defined by Lemma 12.3. The gauge group G Γ acts on both the total and base spaces makingη into a G Γequivariant complex line bundle. Taking orbits, we obtain a complex line bundle η over H 1 (Γ; U (1)) ×M Γ .

Appendix: an adiabatic theorem
Here we formulate and prove an adiabatic theorem for periodic driving. Roughly, it states that for slow enough driving a periodic solution of the master equation exists and is unique, and furthermore, in the adiabatic limit this solution will tend to the Boltzmann distribution taken at the associated normalized driving protocol.
Let us introduce the evolution operator U (t, t 0 ) = U (t, t 0 ; H, τ D ) for 0 ≤ t 0 ≤ t ≤ 1, which is the unique solution to the initial value problem d dt U (t, t 0 ) = τ D U (t, t 0 )H(γ(t)), U (t 0 , t 0 ) = I , where I denotes the identity operator. We remark that U (t, t 0 ) is also called the path-ordered exponential and is sometimes expressed in the notationT exp(τ D t t 0 dt H(γ(t ))) (cf. [L]). Then it is elementary to show that the master equatioṅ has formal solution p(t) = U (t, 0)p(0) .
Proof. We shall use abbreviated notation and write ρ B (t) in place of ρ B (γ(t)). For any solution to the master equation p(t), set ξ(t) := ρ B (t) − p(t) .
Inserting Eq. (33) expression into the master equation and using the fact that the Boltzmann distribution lies in the null space of the master operator, we obtain the first order linear differential equation in ξ, where H(t) is shorthand for H(γ(t)).
Applying U (1, t) to Eq. (34) we get Notice that the left side of the last display is just d/dt(U (1, t)ξ). Integrating both sides we obtain Setting t = 0 we see that U (1, 0)ξ(0) = C. Evaluating at t = 1 and using the fact that U (1, 1) = I yields It is therefore sufficient to show that the operator I − U (1, 0) is invertible, when considered as an operator acting on the invariant subspacẽ C 0 (Γ; R), provided τ D is sufficiently large. Let λ and c be the constants obtained in Lemma 13.2 below. If we set τ 0 := (1/λ) ln(2c), then we have U (1, 0) < 1/2. It follows that I − U (1, 0) is invertible onC 0 (Γ; R).
The last part of the proof of Proposition 13.1 rested on an estimate that appears below. To formulate it we use the norm onC 0 (Γ; R) given by ξ = ξ, ξ where the inner product is the one induced by the standard inner product on C 0 (Γ; R). If A is an operator onC 0 (Γ; R) then we define A := sup ξ =0 Aξ ξ −1 = sup ξ=1 Aξ .
The uniform bound provided by Eq. (37) implies the uniform bound with respect to the standard inner product for some B > 0, which immediately implies the uniform bound U 0 (t, t 0 ) < B.
Proof. It is enough to show that lim τ D →∞ ξ(t) = 0 where ξ(t) is as in the proof of Proposition 13.1. We first show that lim τ D →∞ ξ(0) = 0.