1 3 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 . . . . . . . . 6 1.4 . . . . . . . . . . . . . . . 7
1.4.1 Taylor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.3 δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4.5 . . . . . . . . . . . . . . . . . . . . . 9 1.4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 12 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.9.1 . . . . . . . . . . . . . . . . . . . . . . 21 2.9.2 . . . . . . . . . . . . . . . . . . . . . . 21 2.9.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.9.4 . . . . . . . . . . . . . . . 21
3 23 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1
3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4.1 () . . . . . . . . . . . . . . . . . . . . . 26 3.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 30 4.1 . . . . . . . . . . . . . . . . . . . 30 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.3.2 . . . . . . . . . . . . . . . . . . . . . 35
4.4 . . . . . . . . . . . . . . . . . . . . 36 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 38

1.1 1.1(a)
1 P (x, t) N ∫ ∞
−∞ dxP (x, t) = N (1.1)
[x, x+x] x J(x, t) 1
d
dt
x dyP (y, t) = −J(x+x, t) + J(x, t) (1.2)
1.1(b)x→ 0 ∂P (x, t)
∂t = − ∂
3

Figure 1.2: J(x, t) ∝ −∂P (x, t)/∂x
1.2
J(x, t) = −D∂P (x, t) ∂x
, (1.4)
∂P (x, t)

exp
1.2.1 1 [1] µm 1.3(b)
2 “fluctuation” “thermal fluctuation”
4
3
20 400 419 19 5 20 1905NA1mol [2]1906 6 [3] x
γ dx
a
γ 6πaη. (1.9)
D = RT
γNA . (1.10)
5


x P (x, t) ∂P (x, t)
∂t = D

1.2.2 1.4 p(t) log x(t) ≡ log p(t)log p(t) [0,∞) x(t) (−∞,∞)
dx
1.3 4
7
8 x(t)
6
1.4 1.4.1 Taylor Taylor C∞9 f(x) x0
f(x) = ∞∑ n=0
(x− x0) n

a dxf ′(x)g(x) = [f(x)g(x)]ba −
∫ b
lim x→∞
limx→∞ P (x) = c > 0 ∫ ∞
−∞ dxP (x) = ∞. (1.19)
limx→∞(∂P (x)/∂x) = c = 0limx→∞ P (x) = 0 (1.17)
9
−∞ dxf ′(x)g(x) = −
δ(x) =
∂
∫ ∞
−∞ dxf(x)δ(ax) =
1
−∞ f(x)δ(ax)dx =
δ(ax) = 1
|g′(xn)| . (1.27)
xn g(x) = 0 n n g′(xn) = 0
1.4.4 A x x ∈ [x, x+ dx] P (x)dx . . .
f(x) = ∫ ∞
8
1.4.5 x P (x) y = f(x)11 y q(y)
P (x)dx = q(y)dy (1.29)
−∞ dxf(x)g(x) =
(1) L2 f(x) = g(x)− h(x) (1.32)
∫ ∞
g(x) = h(x) 12
(3) δ f(x) = δ(x− y) (1.34)
g(y) = h(y) (1.35)
∂P (x, t)
11 f(x) 12 L2-

9
δ P0(k) =

∫ ∞
(1.36)
∂P (k, t)
1√ 2π e−ikx0−Dk2t. (1.42)
(4) (1.42)
P (x, t) = 1√ 2π
∫ ∞
exp
2
4Dt
) . (1.45)
13 f f 14a > 0 b 9∫ ∞
−∞ dxe−
215

P (x) = 1√ 2π e−x2/2. (1.48)
y = σx+ µ (1.49)
y Q(y) =
y = x2 yQ(y)
Q(y) = e−
∫ ∞
2.1 v(t) 1
∂
∂t P (v, t) = LP (v, t). (2.1)
P (v(t) = v) ≡ P (v, t)L (2.1) Chapmann-Kolmogorov
(1.7) L
L = D ∂2
dv
1 [9]
12
(a) (b) (c)
Figure 2.1: (a) F ext(t) = y∗δ(t − τ). (b)dv/dt = −v + F ext.(c) τ [9]
v0) = P0(v0) Liouville
∂
Liouville f(v)
df(v(t))
dt = −a(v)df(v(t))
dv . (2.5)
⟨ df(v)
df(v)
dt
dv
dt = −a(v) + F ext, F ext = y∗δ(t− τ). (2.8)
a(v)y∗ τ P (v(0) = v0) = P0(v0) τ y∗ v 2.1
13
Liouville
∂
∂
∂v a(v)P (v, t) + [P (v − y∗, t)− P (v, t)]δ(t− τ). (2.9)
(2.8) Liouville f(v) dt

f(v(t) + y∗)− f(v(t)) (τ ∈ [t, t+ dt]) . (2.10)
δ df(v)
δ f(v)
[f(v + y∗)− f(v)]δ(t− τ) = [f(v(τ − 0) + y∗)− f(v(τ − 0))]δ(t− τ). (2.12)
⟨ df(v)
⟩ ∫ ∞
} P (v, t)∫ ∞
{ ∂
∂v a(v)P (v, t)dt+ [P (v − y∗)− P (v)]δ(t− τ)
} f(v). (2.13)
14
λdt+O(dt2). (2.14)
ξPy∗,λ(t) = ∞∑ i=1
y∗δ(t− ti). (2.15)
dv
a(v) 2.2(b) ( a(v) = v) 2.2(b)
∂P (v, t)
∂
∂v a(v)P (v, t) + λ[P (v − y∗, t)− P (v, t)]. (2.17)
(∂/∂v)a(v)P (v, t) λP (v − y∗, t) −λP (v, t) ξP(t;λ) ≡ ξPy∗=1,λ(t) (2.17) f(v) dt

df(v(t)) =
f(v(t) + b)− f(v(t)) (ti ∈ [t, t+ dt]: = λdt) . (2.18)

−∞ dvf(v)[P (v, t+ dt)− P (v, t)] =
∫ ∞
} P (v, t)∫ ∞
∂v a(v)P (v, t) + λ[P (v − y∗)− P (v)]
} f(v)dt.
(2.19)
∂v a(v)P (v, t) + λ[P (v − y∗)− P (v)]
} f(v) (2.20)
15
(a(v) = 0) [t, t+ dt] dv(t) ≡ v(t+ dt)− v(t)
dv =
0 (ti ∈ [t, t+ dt] : = 1− λdt) . (2.21)

(dv)n =
0 (ti ∈ [t, t+ dt] : = 1− λdt) . (2.22)
(dv)n = λy∗ndt. (2.23)
O(dt)
2.4.1
ξSPy∗,λ(t) ≡ ξPy∗,λ/2(t) + ξP−y∗,λ/2(t). (2.24)
ξPy∗,λ/2(t) ξP−y∗,λ/2(t) 2.3(a)
dv
∂P (v, t)
λ
16
(dv)n =
2.4.2 N k y∗k λk
ξCP y∗,λ(t) ≡
N∑ k=1
ξPy∗k,λk (t). (2.28)
y∗ ≡ (y∗1, . . . , y ∗ N )λ ≡ (λ1, . . . , λN ) (2.28)
dv
∂t =
λk[P (v − y∗k, t)− P (v, t)] (2.30)
(2.17)
y λ(y) [t, t + dt] y ≤ y∗ ≤ y + dy y∗
λ(y)dydt (2.31)
ξCP λ(y)(t) ≡
ξCP λ(y)(t) ≡
(2.35) .
17
2.5 ξSPy∗,λ y∗ → 0 λ → ∞ σ2 ≡ λy∗2 = const. ( 2.4(a, c)) 2
ξGσ2(t) ≡ lim λy∗2=σ2
y∗→0
dv
18
∂t =
λ
= ∂
∂v2n P (v, t). (2.38)
(λy∗2 = σ2 (const.)) y∗ → 0 ∂P (v, t)
∂t =

W (t) ≡ ∫ t
(dW )n =
2.6 v ξ t1, t2, t3
ξ(t1)ξ(t2)v(t3)=v = Cδ(t1 − t2). (2.42)
· · ·v(t)=v v(t) = vC t1, t2, t3, v [5,11] ξm ξCP
λ(y) ξG:
∑ y
∂P (v, t)
19
λ(y). (2.45)
L(t) ≡ ∫ t
2.7 v vv
σ2 → b2(v) λ(y) → λ(y|v) (2.47)
ξGσ2(t) → ξGb2(v)(t), ξCP
∂t =
∫ ∞
−∞ dy[λ(y|v − y)P (v − y, t)− λ(y|v)P (v, t)]. (2.49)
b2(v) v(t) = vλ(y|v) v(t) = v [5] (2.49) v
dv
λ(y|v). (2.50)
2.8 (2.49) (2.49) C∞ ∫ ∞
−∞ dyλ(y|v − y)P (v − y, t) =
∞∑ n=0
3 “ · ”
αn(v) ≡ ∫ ∞
2.9 2.9.1 (2.26)
2.9.2 (2.30)
dv
∂P (v, t)
e−γv2/σ2 (2.58)
dv
dv
dt = v(t)δ(t− 1). (3.3)
v(0) = 1 (3.3) well-definedv(t) δ(t−1)tv(1+t) = vi+1v(1−0) = vi 3.1
• ( 3.1)
vi+1 − vi t
23
Figure 3.1: δ (a) v(t) (b) (b) v v(t) = 1
• ()
• α- (0 ≤ α < 1)
vi+1 − vi t
t ⇐⇒ v(1 + 0) =
3.2 1 (3.4) [0, T ] (0 = s0 < s1 < · · · < sN = T ) ξ(t)N → ∞ ∫ T
0 f(v(t)) · ξ(s)ds =
f(x)L(t) ≡ ∫ t 0 dsξ(s)L(si) ≡ L(si+1) − L(si) =
ξ(si)si|t| ≡ maxi |ti+1 − ti| “ · ” f(v) ξ2 (3.7)
dv
1
dv = −a(v)dt+ b(v) · dL ⇐⇒ v(t) = v(0)− ∫ t
0 a(v(s))ds+
Li vi vi+1 (3.11)
3.3 SDE
dv

v dv(t) ≡ v(t+ dt)− v(t) dv
dv(t) =
−a(v)dt (ti ∈ [t, t+ dt]) . (3.15)
v y = f(v)f(v)
f(v) df(v) ≡ f(v(t+ dt))− f(v(t))
df(v(t)) = f(v(t+ dt))− f(v(t)) =
25
t = ti (3.16) f ′(v) y∗ (3.16)
df(v) =
3.4 ξ(t) = ξG(t) ( L(t) = W (t)) (3.16)
3.4.1 ()
(i) (i)
(ii) (ii) (2.41) ξG ( W ) O(dt) 3
(dW )n =
∫ t
. (3.20)
26
(3.20) (Mean Square Convergence) t→ 0 A B
lim t→0
f(vi)W 2 i −
N−1∑ i=0
]2⟩
2 j −tj)
(W 2
i = o(ti)
lim |t|→0
3.4.2 ()
dv
(3.18) (3.24) (2.36) f(v)
df(v) =
[ −df(v)
df(v) =
dtdW = 0 (3.19) (3.25) (3.26). (3.25) (3.26)
3.4.3 SDE (3.24)SDE (3.24)
∂P (v, t)
df(v)
dt
3.5 3.5.1 (1) f(W )
f(W )
dS = µSdt+ σSdW , W (0) = 0. (3.33)
(3.32)
S(t) = S0 exp
logS(t) = logS0 +
( µ− σ2
M dv
dt = −γv +
G. (4.1)
v ≡ dx/dtM γ kB ≡ R/NA a η
γ = 6πaη (4.2)

3 (4.1)
τp ≡ M
2 3 (4.1)
2010 [13] 2014 [14]
30
τp τo =⇒ γ dx
D = kBT
MSD(t) ≡ (x(t)− x0) 2 = 2Dt (4.7)
6
NA = RT
6πaηD . (4.8)
4.2.1 V
pV = nRT, (4.9)
pV nmolRT NNA ρ
p = ρkBT, kB ≡ R
4 2.8µm [14] τp 50− 150µs [13] 5 [5] Adiabatic elimination 6MSDt

31

Figure 4.1:
4.2.2 v ≡ (vx, vy, vz) f(vx, vy, vz)
f(vx, vy, vz) = (vx)(vy)(vz), (v) ≡ √
m
[ − mv2
2kBT
] (4.11)
vx, vy, vz
f(vx, vy, vz) = (vx)(vy)(vz), (v) ≡ √
1
2mvx (4.13)
4.1t vz S ρSvxt(vx) (4.14)
pSt
kBT
M dV
PSS(V ) =
√ M
32
β =
√ 2γkBT (4.18)
4.3.1 [12, 17, 18] 1 1 4.2 T (v) M m 1 ( x ) V vz e = 1
mvx +MV = mv′x +MV ′, e = −V ′ − v′x
V − vx = 1 (4.19)
V ′ ≡ V − 2m
2M
y ≡ V ′ − V
y = − 2m
V t vx
p(vx|V )tdvx = ρS|V − vx|t(vx)dvx (4.22)
8 [15, 16]
33
p(vx|V ) y λ(y|V )
λ(y|V )dy = p(vx|V )dvx (4.23)
⇐⇒λ(y|V ) = ρS(m+M)2
∂P (V, t)
∫ ∞
−∞ dy[λ(y|V − y)P (V − y, t)− λ(y|V )P (V, t)] (4.25)
⇐⇒∂P (V, t)
∫ ∞
−∞ dvx|vx − V |{P (V ′, t)(v′x)− P (V, t)(vx)}. (4.26)
dV
PSS(V ) = lim t→∞
P (V, t) =
Ffriction(V ) −γV +O(V 2) (4.29)

9 [19–23] 10Carlo Cercignani
[24]
M m =⇒ ε ≡ 2m
m+M → 0. (4.30)
∂P (V, t)
∫ ∞
αn(V ) = ρS
Y = y/ε. ε
∂P (V, t)
∫ ∞
V
A1(V ) =
A1(0) = 0
ε→ 0 ??(a) (Z ≡ V/
√ ε)ε→ 0
??(b) (τ ≡ εt) p(Z, τ) ≡
√ εP (
√ kBT
∂τ =
[ γ
2m
∂t =
[ γ
M
λ(y|v)
Y = εy (4.42)

λ′ε(Y |V )dY = λ(y|V )dy ⇐⇒ λ′ε(Y |V ) = 1
ε λ
∂t =
−∞ dY
] =
( An(V ) ≡
A (0) 1 (0) = 0, A
(1) 1 (0) < 0, A(0)
n (0) > 0 (for n = 2) (4.46)
ε→ 0 (4.46) [6]
36
(1) f(vx, vy, vz) 2
• f(vx, vy, vz)v ≡ |v|
f(vx, vy, vz) = F (v2) (4.47)
•
2 z). (4.48)
] , (4.50)
m
−∞ δ(x− y)f(x) = f(y) (5.1)
(2)
∂t =
∂x
∂
lim
x→±∞
∂
log P (k, t) = −Dk2t+ C(k), (5.5)
C(k) k
2 log(2π) = C(k) (5.6)
1√ 2π e−ikx0−Dtk2 . (5.7)
(4) ∫ ∞
∫ ∞
1√ 2π
P (x)dx = Q(y)dy, dx
dv
f(v) df(v(t)) ≡ f(v(t + dt)) − f(v(t)) 3
df(v) =
−a(v)df(v)dv (1− λdt )
2 [f(v − y∗)− f(v)] − (1− λdt)
⟨ a(v)
df(v)
2 [f(v − y∗)− f(v)] −
df(v)
dv
] + o(dt)
λ
∂
y∗,λ
df(v) =
{ f(v + y∗k)− f(v) (λkdt, k = 1, 2, . . . , N )
−a(v)df(v)dv (1− λdt ) . (5.21)
λ ≡ ∑N
df(v) = N∑ k=1
⟨ a(v)
df(v)
df(v)
dv
dv
] + o(dt)
∂v a(v)P (v, t)
λ(y|v) (5.24)
⇐⇒ ∂P (v, t)
∫ ∞
−∞ dy [λ(y|v − y)P (v − y, t)− λ(y|v)P (v, t)]
(5.25)
a(v) = 0, b(v) = σ, λ(y|v) = 0 (5.26)
∂P (v, t)
∂t = σ2
41
2.9.4 (1) (5.24)
a(v) = γv, b(v) = σ, λ(y|v) = 0 (5.29)
∂P (v, t)
∂t =
[ γ ∂
dv v +
σ2
2
d
dv
dv logPSS(v) = −2γ
σ2 v (5.35)
logPSS(v) = − γ
[ − γ
W
dW 2 = dt, dWdt = 0, dWn = 0, (n ≥ 3) (5.39)

df(W ) = ∞∑ n=0
df(W , t) = ∂f(W )
2
S(t) = f(W , t) 1 S(t) = f(W , t)
4 4.5.1 (1)
f(vx, vy, vz) = F (v2x + v2y + v2z) = ψ(v2x)ψ(v 2 y)ψ(v
2 y) (5.44)
F (v2x) = ψ(v2x)a 2, a ≡ ψ(0) = 0. (5.45)
43
z = 0 aψ(x+ y) = ψ(x)ψ(y). (5.47)
y 1
4.5.2
[2] A. Einstein, Ann. Phys. (Leipzig) 17, 549 (1905).
[3] P. Langevin, Comptes Rendues 146, 530 (1906).
[4] F. Black and M. Scholes, Journal of Political Economy 81, 637 (1973).
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