Schrödinger Equations and Diffusion Theory || The Schrödinger Equation can be a Boltzmann Equation

Chapter VIII

The Schrödinger Equation canbe a Boltzmann Equation

8.1. Large Deviations1

Let us consider the Schrödinger process {Xt, Q} determined by the triplet{P(s, x)

c , μa, μb} (cf. Chapter 5), namely Q is Csiszar's projection on thesubset Aa,b (see (8.4) below) of the renormalized process P, or a diffusionprocess constructed by the method of Chapter 6 for a given function φ(t, x).In the latter case we take the creation and killing c = - Lφ/φ induced by

φ(t, x), and another function φ(a, x) such that μa(x) = φ(a, x)φ(a, x), andmoreover assume the admissibility condition (5.12) so that Q = Qμa is theCsiszar projection of P. This case occurs, if a solution ψ = eR + i S of theSchrödinger equation is given beforehand and if we define φ = eR + S and

φ = eR - S in terms of R and S of ψ = eR + i S.

Our main theme in this chapter is this:

Find a sequence Q(n) of systems of interacting n-diffusion processessuch that

Q(n) converges to Q ∞, the infinite product of Q, as n tends to infinity,

a precise definition of which will be given in (8.2). The meaning of theconvergence is this: If the total number n of particles in the system Q(n)

tends to infinity, then the interacting n-particles in the system becomeasymptotically independent of each other and indistinguishable, namely everyparticle will move independently according to the same probability law Q.

1 Cf. Chapter 11, in which a main theorem on large deviations is shown

207, , DOI 10.1007/978-3-0348-© Springer Basel M. Nagasawa chrödinger Equations and Diffusion Theory S 0560-5_8,

1993

208 Chapter VIII: The Schrödinger Equation can be a Boltzmann Equation

To be precise we consider a system {Ωn, Q(n, k)} of interacting n-diffusion particles, where we allow one more parameter k, which will beexplained soon in the following.

The drift coefficient bin,k(t, x) = bi

n,k(t, x, Ln(x)), x = (x1, ... , xn) ∈ (Rd)n,

i = 1, 2, ... , n, of the interacting diffusion processes (X 1, . . . , X n) must be

found so that the propagation of chaos holds. The coefficient bin,k(t, x)

represents the interaction of the i-th particle X i with the other particles:This means, the i-th particle X i obeys a stochastic differential equation

Xti = Xa

i + bin,k(r, Xr)

a

t

dr + σa

t

(r, Xr).dBri ,

where X t = (Xt1, . . . , Xt

n) and Bti, i = 1, 2, ... , n , denote mutually

independent d-dimensional Brownian motions. The process X t depends onn and k, which are not indicated to avoid complication of notations.

The empirical distribution of the system of interacting diffusionprocesses (X1, ... , Xn) is defined by

(8.1) Ln(ω ) = 1n δω i∑i = 1

n

, ω = (ω1, ... , ωn) ∈ Ωn.

The standard definition of the propagation of chaos is given as follows:For the system {Ωn, Q(n, k)} of Markov processes, the propagation ofchaos holds with the limiting distribution Q as n → ∞ and k → ∞, if it hasthe following two properties

Q(n, k) converges weakly to the infinite product of Q,(8.2)

Q(n, k)[Ln] converges weakly to Q,

as n → ∞ and k → ∞, where Ln is the empirical distribution defined in (8.1).

This standard definition will be modified (cf. (8.17), (8.18)) to besuitable with a new formulation of the propagation of chaos in terms of largedeviations.

In our context the limiting process Q in (8.2) must be the Schrödingerprocess (Csiszar's projection of the renormalized measure P) determined by

8.1 The Large Deviations 209

the triplet {P(s, x)c , μa, μb}. Thus, our problem is reduced to constructing a

sequence {Q(n, k): n , k = 1, 2, ...} of systems of interacting diffusionprocesses which satisfy (8.2). This is an inverse problem of the propagationof chaos.

To solve this problem we apply theorems on large deviations due toCsiszar (1984), in particular his asymptotic quasi-independence of a sequence{Q(n, k): n, k = 1, 2, ...} of diffusion processes (probability measures) inconnection with the Sanov property.

Although the statement itself of the Sanov property (see, Theorem 10.1)does not contain any topological concept, to prove it we need a topologicalstructure on the space M1(Ω) of probability measures on Ω, but we need notopological structure on Ω. In fact, Csiszar (1984) defines an intrinsictopology of M1(Ω) which depends only on the measurable structure of Ω.

Csiszar's τ0-topology is given in terms of the basic neighbourhoods ofP ∈ M1(Ω) :

(8.3) U(P, ε, Pk) = {R∈M1(Ω) : R(Ωi) - P(Ωi) < ε, for i = 1, 2, ... , k,

and R << P on σ (Pk), i.e., R(Ωi) = 0, if P(Ωi) = 0},

where

Pk = Pk(Ω) = {Ω1, ... , Ωk}, k = 1, 2, ... , are finite measurable

partitions of Ω,

σ(Pk) is the σ-field generated by the partition Pk.

Thus the neighbourhood U(P, ε, Pk) depends only on a finite measurablepartition Pk and ε.

Let the subset Aa,b of M1(Ω) be defined by (5.11), i.e.,

(8.4) Aa,b = {P∈M1(Ω) : P°Xr-1 = μr, for r = a, b},

where μr ∈ M1(Rd), r = a, b, are fixed. We assume that the set Aa,b isadmissible for the reference process P (cf. (5.12)), so that the Csiszarprojection Q of P on the set Aa,b exists.


The set Aa,b has played an important role in discussing variationalprinciple in Chapter 5. However, the lateral conditions defining the subsetAa,b is too strict for an application of Sanov's theorem. One of the troubleswith Aa,b is this: the set of inner points of Aa,b in Csiszar's τ0-topology isan empty set and hence the lower bound in Sanov's theorem2 becomesuseless. Therefore, loosening the lateral conditions at the initial and terminaltimes t = a, b of the subset Aa,b , we approximate it by a family of subsetsA(ε, k) of M1(Ω) defined by

(8.5) A(ε, k) = {P ∈M1(Ω) : P[Xr ∈Bi] - μr(Bi) ≤ ε2k

, for ∀ Bi ∈ Pk(Rd),

and P°Xr-1 << P°Xr

-1 on σ(Pk(Rd)), r = a, b},

for ε > 0 and k = 1, 2, ... , where P is a prescribed reference process, andPk(Rd) = {B1, ... , B2k}, k = 1, 2, ... , are finite measurable partitions of Rd

such that Pk+1(Rd) is a refinement of Pk(Rd) :

σ(Pk(Rd)) ⊂ σ(Pk+1(Rd)) and σ(Pk(Rd)) ↑ σ(Rd), as k ↑ ∞.

The dependence on ε > 0 of the subset A(ε, k) is not substantial but technical.In the proof of Theorem 8.1 we will use it in (8.11).

Let P = Pn be the n-fold product of P. The conditional probability

P(n, k) of P is defined, for a fixed ε > 0, by

(8.6) P(n, k)[ . ] = PA(ε, k)(n)

[ . ] = P[ . | Ln ∈ A(ε, k)],

with which the motion of the n-independent particles governed by P iscontrolled so that the empirical distribution Ln becomes not very differentfrom the prescribed probability distributions μa and μb at the initial andterminal times t = a, b in the sense of (8.5). Therefore, although it isclear because of the conditioning, it should be emphasized that the process{(X1, ... , Xn), P(n, k)} with the conditional probability P(n, k) does not have theMarkov property and (X1, ... , Xn) is not independent any more.

Then we will formulate the fundamental assertions of the theory of largedeviations in a generalized form as an "approximate Sanov property" and(approximately) asymptotic quasi-independence.

2 Cf. Lemma 11.5 in Chapter 11


Theorem 8.1. (Aebi-Nagasawa (1992)) Assume the conditions ofadmissibility (5.12) and integrability (5.19).

(i) Let Aa,b be the subset of M1(Ω) defined in (8.4) and A(ε, k) be theone defined in (8.5), for a fixed ε > 0. Then

(8.7) Aa,b = A(ε, k)∩k∈N

,

and the "Approximate Sanov Property" holds for the subset Aa,b :

(8.8) limk → ∞

limn → ∞

1n log P[Ln ∈ A(ε, k)] = - inf

P∈Aa,b

H(P | P)

= - H(Q | P ),

where the infimum is attained by Csiszar's projection Q (the Schrödingerprocess determined by the triplet {P(s, x)

c , μa, μb}) of the reference processP (the renormalized process). In other words, the probability that therare event {Ln ∈ A(ε, k)} occurs is given by

P[Ln ∈ A(ε, k)] ≈ e-n H(Q | P), as n → ∞ and k → ∞.

(ii) The conditional process {(X1, . . . , Xn), P(n, k)} is (approximately)asymptotically quasi-independent with the limiting distribution Q; namely,

(8.9) limk → ∞

limn → ∞

1n H(P(n, k) | (Qk)

n) = 0,

and Qk converges to Q in entropy as k tends to infinity, where Qk denotesthe Csiszar projection of the reference process P on the set A(ε, k).

Proof. A key step of the proof is to show the inclusion

(8.10) A°(ε1, k) ⊂ A(ε1, k) ⊂ A°(ε2, k) ⊂ A(ε2, k),

where 0 < ε1 < ε2 and A°(ε, k) denotes the interior of A(ε, k) with respect toCsiszar's τ0-topology.

Non-trivial inclusion in (8.10) is the second one which follows from


Lemma 8.1. For any ε0 > 0

(8.11) ∪0 < ε < ε0

A(ε, k) ⊂ A°(ε0, k).

Proof. Let P0 ∈ A(ε, k)∪0 < ε < ε0

, i.e., P0 ∈ A(ε1, k), for 0 < ε1 < ε0. For

each Bi ∈Pk(Rd) define Ωir by

Ωir = {ω : Xr(ω ) ∈ Bi}, for r = a, b,

and a finite measurable partition P of Ω by

P = P(Ω) = {Ωia ∩ Ωj

b : i, j = 1, 2, ... , k}.

Let us take P ∈ U(P0, ε2, P). Then P << P0 on σ(P), and moreoverP0°Xr

-1 << P°Xr-1 and hence P°Xr

-1 << P°Xr-1 on σ(Pk(Rd)), for r = a , b.

For ∀ Bi ∈ Pk(Rd) we have

| P[Xr ∈Bi] - qr(Bi) | ≤ | P[Ωir] - Po[Ωi

r] | + | P0[Xr ∈Bi] - qr(Bi) |

≤ kε2 + ε1

2k, for r = a, b.

Therefore, if we choose ε2 > 0 such that kε2 + ε1

2k ≤ ε0

2k, then P ∈ A(ε0, k),

i.e., U(P0, ε2, P) ⊂ A(ε0, k), which implies (8.11), and completes the proof.

We denote for simplicity

H(A | P) = infP∈A

H(P | P).

Then (8.10) combined with (8.7) yields

(8.12) limk → ∞

H(A°(ε, k) | P) = limk → ∞

H(A(ε, k) | P) = H(Q | P),

where Q is the Csiszar projection on the subset Aa,b of P.

Since Q ∈ A(ε, k) and Qk is the Csiszar projection on the set A(ε, k) ofthe reference process P, Csiszar's inequality (10.12) implies


H(Q | P) - H(Qk | P) ≥ H(Q | Qk),

the right-hand side of which vanishes because of (8.12), and hence we have

(8.13) limk → ∞

H(Q | Qk) = 0;

namely, Qk converges to Q in entropy and hence in variation (cf. Lemma10.2).

To apply a theorem of Csiszar on the Sanov property (cf. Chapter 11) tothe subset A(ε, k) we need

Lemma 8.2. The subset A(ε, k) ⊂ M1(Ω) is completely convex 3

and also variation closed.

Proof. Let (Λ, A, μ) be an arbitrary probability space and η(λ, B) be

a probability kernel on Λ × σ(Ω) such that η(λ, . ) ∈ A(ε, k), for ∀λ ∈Λ.Then, for any Bi ∈Pk(Rd),

| μη[Xr ∈Bi] - qr(Bi) | ≤ | η(λ,{Xr ∈ Bi}) - qr(Bi) | μ(dλ)

≤ ε2k

, for r = a, b,

where

μη( . ) = μ(dλ)η(λ, . ) .

Moreover, it is clear that μη°Xr-1 << P°Xr

-1 on σ(Pk(Rd)), for r = a , b .

Therefore, μη ∈ A(ε, k), i.e., the set A(ε, k) is completely convex.

To show that the set A(ε, k) is variation closed, let {Pn} ⊂ A(ε, k) be asequence which converges to P ∈ M1(Ω) in variation, i.e.,

|| Pn - P ||Var = dPn

dP - dP

dP dP → 0, as n → ∞,

with respect to P. Then, for any Bi ∈Pk(Rd) and r = a, b,

3 Cf. Section 2 of Chapter 11


P[Xr ∈ Bi] - qr(Bi) ≤ P[Xr ∈ Bi] - Pn[Xr ∈ Bi] + Pn[Xr ∈ Bi] - qr(Bi) ,

where the second term on the right-hand side is evaluated as

| Pn[Xr ∈Bi] - qr(Bi) | ≤ ε2k

,

and the first term vanishes, namely

| P[Xr ∈Bi] - Pn[Xr ∈Bi] | ≤ || Pn - P ||Var → 0, as n → ∞.

Combining these inequalities, we have

| P[Xr ∈Bi] - qr(Bi) | ≤ ε2k

,

and hence P ∈ A(ε, k), i.e., the set A(ε, k) is variation closed, whichcompletes the proof.

Therefore, applying a theorem of Csiszar (cf. Theorem 11.1) to the setA(ε, k), we have the Sanov property of the subset A(ε, k):

(8.14) - H(A°(ε, k) | P) ≤ 1nliminf

n → ∞ log P[Ln ∈A(ε, k)]

≤ 1nlimsup

n → ∞ log P[Ln ∈A(ε, k)] ≤ - H(A(ε, k) | P),

namely

1n

limn → ∞

log P[Ln ∈A(ε, k)] = - H(Qk | P),

where H(A(ε, k ) | P) = H(Qk | P) with the Csiszar projection Qk on the

subset A(ε, k) of P. The inequality (8.14), together with (8.12), yields theapproximate Sanov property (8.8).

The second assertion on the asymptotic quasi-independence (8.9) of P(n, k)

follows from Csiszar's inequality (cf. equation (11.10) in Theorem 11.1)

(8.15) 0 ≤ 1n H(P(n, k)

| (Qk)n) ≤ - H(A(ε, k) | P) - 1

n log P[Ln ∈ A(ε, k)],

together with the approximate Sanov property (8.8); namely the right-hand

8.2 The Propagation of Chaos in Terms of Large Deviations 215

side of (8.15) vanishes as n → ∞ and k → ∞. This, combined with (8.13),proves the second assertion on the asymptotic quasi-independence of P(n, k)

and completes the proof of the theorem.

8.2. The Propagation of Chaos in Terms of Large Deviations

The conditional process {(X1, . . . , Xn) , P(n, k)} in Theorem 8.1 is notMarkovian as we have remarked already. However, we can find a uniqueMarkovian modification Q(n, k) of P(n, k), applying Theorem 5.4, theentropy condition of which is satisfied automatically in this case as will beseen.

Let P and P0 be probability measures on the space of right-continuouspaths on Rd with t ∈ [a, b]. Then P0 is called a Markovian modificationof P , if P0 is Markovian and the marginal distributions at time t of P andP0 coincide for ∀ t ∈ [a, b].

Theorem 8.2. For the conditional process P(n, k) (cf. Definition (8.1))there exists a unique Markovian modification Q(n, k) such that

(8.16) H(Q(n, k) | (Qk)n) ≤ H(P(n, k) | (Qk)

n).

Proof. We apply Theorem 5.4 for P = P(n, k) and Q = (Qk)n, where Qk

is the Csiszar projection on the set A(ε, k) of P, setting μt = P(n, k)°Xt

-1 in the

set A defined at (5.43). Because H(P(n, k) | (Qk)n) < ∞ is clear from (8.9),

the admissibility condition of the set A for Q = (Qk)n is satisfied, and hence

we have the Markovian modification Q(n, k), which is the Csiszar projectionof Q = (Qk)

n on the set A. This completes the proof.

We define "the propagation of chaos" in a stronger form to be fit forour treatment in terms of large deviations as follows:

For a sequence of systems {(X1, ... , Xn), Q(n, k)} of interacting diffusionprocesses, the propagation of chaos in entropy holds with a limitingdistribution Q as n → ∞ and k → ∞, if for arbitrary but fixed m ≥ 1 and forthe marginal distribution Qm

(n, k) of Q(n, k) on Ωm


Qm(n, k) converges to the m-fold product Q m of Q in entropy,

(8.17) Q(n, k)[Ln] converges to the Q in entropy,

as n → ∞ and then k → ∞.

Each convergence in (8.17) consists of two steps. In the first propertythis means that H(Qm

(n, k) | (Qk)

m) → 0 as n → ∞ and then H(Q m | (Qk)

m) → 0 ask → ∞ . The second property should be understood similarly.

Besides the propagation of chaos in entropy (8.17), we can formulatethe propagation of chaos in variation as follows:

Qm(n, k) converges to the m-fold product Q m of Q in variation,

(8.18) Q(n, k)[Ln] converges to the Q in variation,

as n → ∞ and then k → ∞.

Then we have

Lemma 8.3. (i) The propagation of chaos in entropy (8.17) impliesthe propagation of chaos in variation (8.18).

(ii) The propagation of chaos in variation (8.18) implies the (weak)propagation of chaos (8.2).

Proof. Since the second property of (8.18) can be shown similarly, weprove only the first one. If (8.17) holds, then, with the help of Lemma 10.2,we have

{|| Qm(n, k) - (Qk)

m ||Var}

2 ≤ 2 H(Qm(n, k)

| (Qk)m) → 0,

{|| Q m - (Qk)m

||Var}2 ≤ 2 H(Q m

| (Qk)m) → 0,

as n → ∞ and k → ∞, and hence

|| Qm(n, k) - Q m

||Var ≤ || Qm(n, k) - (Qk)

m ||Var + || (Qk)

m - Q m ||Var → 0,

as n → ∞ and k → ∞. The propagation of chaos (8.2) clearly follows fromthe propagation of chaos in variation (8.18). This completes the proof.

217

Our main theorem in this section states:

Theorem 8.3. (Aebi-Nagasawa (1992)) Assume the admissibilitycondition (5.14) and the integrability condition (5.19).

Let {(X1, ... , Xn), Q(n, k)} be the system of interacting diffusion processwith Q(n, k) which is the Markovian modification of P(n, k) given inTheorem 8.2 such that

H(Q(n, k) | (Qk)n) ≤ H(P(n, k) | (Qk)

n).

Then:(i) The Markovian modification {(X1, ... , Xn), Q(n, k)} is a system of

interacting diffusion processes with the Markovian drift coefficientbn, k(t, x) (= interaction), x = (x1, ... , xn) ∈ (Rd)

n, such that

(8.19) bn, k(t, x) ={bin, k(t, x, Ln(x)) : i = 1, 2, ... , n},

where bin,k is the drift vector of Xi under (Qk)

n.

(ii) The approximate Sanov property

limk → ∞

limn → ∞

1n

log P[Ln ∈ A(ε, k)] = - infP ∈ Aa,b

H(P | P)

= - H(Q | P )

holds, where the infimum is attained by the Schrödinger process Q(Csiszar's projection) determined by {P(s, x)

c , μa, μb}. In other words, theprobability that the rare event {Ln ∈ A(ε, k)} occurs is given by

P[Ln ∈ A(ε, k)] ≈ e- n H(Q | P), as n → ∞ and k → ∞ .

(iii) The Markovian modification Q(n, k) is asymptotically quasi-independent with the Schrödinger process Q as limiting distribution;

(8.20) limk → ∞

limn → ∞

1n H(Q(n, k) | (Qk)

n) = 0,

and Qk converges to Q in entropy as k → ∞.

8.2 The Propagation of Chaos in Terms of Large Deviations


(iv) Moreover, the propagation of chaos in entropy holds for theMarkovian modification {(X1, ... , Xn), Q(n, k)} with the Schrödingerprocess Q as limiting distribution, when n → ∞ and k → ∞.

Proof . Theorem 8.2 implies the existence of the Markovianmodification Q(n, k). To show assertion (i) we first note that the modificationQ(n, k) is absolutely continuous with respect to (Qk)

n and moreover Qk << P.There exists, consequently, a measurable (vector) function bn, k(t, x) and aBrownian motion {Bt, (Qk)

n} such that

Xti = Xa

i + bin,k(r, Xr)

a

t

dr + σa

t

(r, Xr).dBri ,

by the generalized Maruyama-Girsanov theorem (cf. Liptser-Shiriyayev(1977)). Assertion (ii) has been shown in Theorem 8.1. In assertion (iii) theasymptotic quasi-independence of Q(n, k), namely convergence (8.20) followsfrom (8.9) with the help of (8.16).

For assertion (iv) we note first that the marginal distributions of the

measure P(n, k) on product spaces Ωmr + 1 × ... × Ωm(r + 1), r = 0, 1, 2, ... , are

identically equal to Pm(n, k). Therefore, the marginal distribution Qm

(n, k) ofQ(n, k) on Ωm converges to (Qk)

m in entropy as n → ∞ by Lemma 11.3, since

(8.21) H(Qm(n, k)

| (Qk)m) ≤ H(Pm

(n, k) | (Qk)

m),

and then (Qk)m converges to Q m in entropy as k → ∞ by (8.13). Therefore,

the first property of (8.17) of the propagation of chaos in entropy holds.

It remains to show the second property of (8.17) of the propagation ofchaos in entropy. Since Q(n, k) is the Markovian modification of P(n, k), it issufficient to show that P(n, k)[Ln] converges in entropy to Q. Let us denote

by PΩ(n, k) the marginal distribution of P(n, k) on Ω. Then we have

(8.22) P(n, k)[Ln(B)] = 1n P(n, k)∑i = 1

n

[1B(ωi)] = PΩ(n, k)[B].

Therefore, an application of Lemma 11.3 yields that P(n, k)[Ln] converges toQk in entropy as n → ∞, and then Qk converges to Q in entropy as k → ∞by (8.13). This completes the proof.

8.3 Statistical Mechanics for Schrödinger Equations 219

The assertion (iv) of Theorem 8.3 guarantees the existence of a sequenceof systems of interacting diffusion processes which we have been lookingfor; namely, we can claim that the Schrödinger process {Xt, Q} describes themovement of a "typical" particle of a system of interacting diffusionparticles, when the size n of the system is sufficiently large.

8.3. Statistical Mechanics for Schrödinger Equations

We summarize the results as a statistical mechanics for a sequence of systemsof interacting diffusion processes:4

Let {μ t, t ∈ [a, b]}, be a flow of non-negative distribution densities (inparticular μt = | ψt |

2, where ψt, t ∈ [a, b], a flow of complex valued functionswith the normalization |ψt(x)|2dx = 1). Instead of the flow μ t we can beginwith a diffusion equation with creation and killing c(t, x) or a Schrödingerequation with a potential V(t, x), which are equivalent with each other aswe have shown in Chapter 4.

(First step) There exists, based on the duality relation (time reversal), adiffusion (Schrödinger) process {Xt , Q} such that its distribution densitycoincides with the given flow μ t (say, | ψt |

2) and Schrödinger's factorization

μ t = φ tφ t holds, where μt dx = Q °Xt-1, namely it is the distribution of the

diffusion (Schrödinger) process (say, of a single particle!).

(Second step) Moreover, there exists a sequence {(Xt1, ... , Xt

n), Q(n, k)}of systems of interacting diffusion processes, such that for the Q(n, k) thepropagation of chaos in entropy holds, especially Q(n, k)[Ln] which coincideswith the marginal distribution of Q(n, k) converges in entropy to theSchrödinger process Q as n → ∞ and k → ∞ in this order. Namely, wecan regard μt (say, ψψ !) as the spatial statistical distribution density ofinfinitely many interacting diffusion particles.

In our statistical theory the flow, the Schrödinger equation, and thediffusion equations are macroscopic descriptions of a system ofinteracting particles; the diffusion process {Xt, Q} gives an intermediatedescription of the movement of a "typical particle" in the system; the systemof interacting diffusion processes {(Xt

1, ... , Xtn), Q(n, k)} is a microscopic

4 Cf. Nagasawa (1980, 88, 90)


description. Hence we can complete the diagram in Proposition 4.2 asfollows

L 2-theory Transformation

⇓ ⇓Algebraic Theory ⇔ Wave Theory ⇔ Diffusion Theory

Commutation Schrödinger Time reversal (duality)relation equation of diffusion equation

the propagation of chaos ⇒ ⇑

Statistical Mechanics

Systems of interactingdiffusion particles

The result we have obtained for the second step connecting theintermediate and microscopic is not quite satisfactory, because assertion (i)of Theorem 8.3 gives no explicit form of interaction between diffusionparticles. Nonetheless I would like to claim

Proposit ion 8.1. The Schrödinger equation is a "Boltzmannequation" for a system of interacting particles (= the system of interactingdiffusion processes {(X1, . . . , Xn), Q(n, k)} in Theorem 8.3), as n → ∞ andk → ∞.

I have said "Boltzmann" for the Schrödinger equation here, placing it inthe context of the propagation of chaos as formulated in (8.17) and (8.18),and taking into account the equivalence of Schrödinger and diffusionequations established in Chapter 4. However, if we want to keep a closeanalogy with classical statistical mechanics, it would be relevant to call thedistribution Q(n, k) of the system of n -particles "the microcanonicalensemble (or distribution)" (cf. (8.18)) of, say, "Schrödinger gas", andthe distribution Q itself of the diffusion process the "Gibbs measure". Infact, in the simplest time-independent (reversible) case, the distribution

density μ of Q°Xt-1 is represented as

μ(x) = eU(x),

8.4 Some Comments 221

if the drift coefficient is given by a potential function U(x)

a(x) = 12

σTσ ∇U(x),

as Kolmogoroff had already observed in 1937. Moreover, we can keep afurther analogy with classical statistical mechanics; Ω = the phase space, P =the "normalized" Liouville measure, and so on (see., e.g. Lanford (1973)).Cf. Nagasawa (1990) and also Aebi-Nagasawa (1992).

8.4. Some Comments

In the last chapter entitled "Une analogie entre la mécanique ondulatoire etquelques problèmes de probabilités en physique classique" of Schrödinger(1932), he begins the first paragraph with

"Le sujet que je vais aborder maintenant n'est pas intimement lié auxquestions dont il s'est agi dans les chapitres précédents. Tout d'abordvous aurez l'impression de choses qui ne sont pas du tout liées. Il s'agitd'un problème classique: problème de probabilités dans la théorie dumouvement brownien. Mais en fin de compte, il ressortira une analogieavec la mécanique ondulatoire, qui fut si frappante pour moi lorsque jel'eus trouvée, qu'il m'est difficile de la croire purement accidentelle."

Then he justifies his formulation of Brownian motions given already inSchrödinger (1931) in two ways. One is based on Eddington's interpretation(1928) of the factorization μ t = ψtψt in quantum theory:

"The ψψ* is obtained by introducing two symmetrical systems of ψwaves travelling in opposite directions in time; one of these mustpresumably correspond to probable inference from what is known (or isstated) to have been the condition at a later time."5

Schrödinger's factorization μ t = φ tφ t of distribution densities is exactlyits real-valued counterpart, which is the product of forward and backwardpredictions φ t and φ t, respectively. Namely, what we do is the prediction ofintermediate states, given a pair of data, one at a starting time in the pastand another one at a terminal time in the future (but, as we have remarked,

5 Quoted from Schrödinger (1932). Here ψ* denotes the complex conjugate of ψ


this is the description in the "imaginary or fictitious" evolution). Let us callthis "Eddington-Schrödinger's prediction". This point has been fullydiscussed already in chapters 3, 4, and 5.

Schrödinger's second justification of his formulation of Brownianmotions is essentially in the contexts of "large deviations"; namely, if theyare placed in ordinary diffusion theory, these processes occur as very rareevents, and hence they should (and can) be considered in terms of largedeviations. This was realized by Föllmer (1988), in which he appliedSanov's theorem to independent Brownian motions to mathematizeSchrödinger's idea. Influenced by Föllmer (1988), Dawson-Gorostitza-Wakolbinger (1990) treated the problem in the case of bounded c(t, x) (i.e.,φ(t, x) has no zeros). Theorems 8.1 and 8.3 generalize their results as"approximate Sanov property (8.8)" for unbounded c(t, x) which isindispensable in quantum mechanics.

As a matter of fact there were two obstacles in treating large deviationsfor Schrödinger equations. The first one is this; the subset Aa,b defined in(8.4) has no inner point with respect to Csiszar's τ0-topology, so that thelower bounds of the Sanov property becomes useless. The second one is ; theempirical distribution Ln(ω) defined in (8.1) does not belong to the subsetAa,b, which makes the standard Sanov assertion meaningless. We have seenthat a possible formulation to overcome these difficulties is the approximateSanov Property (8.8) given in Theorem 8.1. A similar way of taking thelimit doubly in a prearranged order as in (8.8) is well known in statisticalmechanics (cf. Lanford (1973)).

We have already remarked in Chapter 6 that a natural way ofinterpreting Schrödinger equations is to consider them as statisticalequations, especially when segregation occurs, namely, as a "Boltzmannequation" of a system of interacting diffusion processes, and a limit theoremof this type has been shown in the framework of the propagation of chaos insome special cases in Theorems 7.1 and 7.3. However, as we have shown, ifwe revise the assertion on asymptotic quasi-independence in Theorem 8.1properly, we can prove the existence of systems of interacting diffusionprocesses in general cases in higher dimensions as stated in Theorem 8.3.Therefore, my attempt to find out a statistical mechanical structure for theSchrödinger equation can be considered as a natural step along the line ofSchrödinger's second justification in terms of large deviations.

Summarizing, we have placed quantum mechanics at its natural positionin the long historical evolution of statistical mechanics originated by LudwigBoltzmann.

Schrödinger Equations and Diffusion Theory || The Schrödinger Equation can be a Boltzmann Equation

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