Reversible Irreversible Engineering
Electric network for non-reversible Markov
chains
Joint work with Áron Folly
Márton Balázs
University of Bristol
Random walks on graphs and potential theory
University of Warwick, 20th May 2015.
Reversible Irreversible Engineering
Reversible chains and resistors
Reducing a network
Thomson, Dirichlet principles
Monotonicity, transience, recurrence
Irreversible chains and electric networks
The part
From network to chain
From chain to network
Effective resistance
What works
The electric network
Reducing the network
Nonmonotonicity
Dirichlet principle
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Reversible chains and resistors
Irreducible Markov chain: on Ω, a 6= b, x ∈ Ω,
hx : = Px {τa < τb }
(τ is the hitting time)
is harmonic:
hx =
X
Pxy hy ,
ha = 1,
hb = 0.
y
a
x
y
b
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Reversible chains and resistors
Irreducible Markov chain: on Ω, a 6= b, x ∈ Ω,
hx : = Px {τa < τb }
(τ is the hitting time)
is harmonic:
hx =
X
Pxy hy ,
ha = 1,
hb = 0.
y
a
1V
a
y
x
x
Rax
Rxy
y
b
Ryb
b
0V
i
Electric resistor network: the voltage u is harmonic (C = 1/R):
ux =
X
y
C
P xy · uy ;
z Cxz
ua = 1,
ub = 0.
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Reversible chains and resistors
Irreducible Markov chain: on Ω, a 6= b, x ∈ Ω,
hx : = Px {τa < τb }
(τ is the hitting time)
is harmonic:
hx =
X
Pxy hy ,
ha = 1,
hb = 0.
y
a
1V
a
y
x
x
Rax
Rxy
y
b
Ryb
b
0V
i
Electric resistor network: the voltage u is harmonic (C = 1/R):
ux =
X
y
C
P xy · uy ;
z Cxz
ua = 1,
ub = 0.
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Reversible chains and resistors
Thus,
Cxy
Pxy = P
z Cxz
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Reversible chains and resistors
Thus,
Cxy
Cxy
Pxy = P
=:
.
Cx
z Cxz
Pxy = Cxy /Cx
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Reversible chains and resistors
Thus,
Cxy
Cxy
=:
Pxy = P
.
Cx
z Cxz
Stationary distribtuion:
µx =
X
µy Pyx =
y
Cx =
y
X
y
Pxy = Cxy /Cx
X
Cy
Cxy
Cy
µy
Cxy
Cy
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Reversible chains and resistors
Thus,
Cxy
Cxy
=:
Pxy = P
.
Cx
z Cxz
Stationary distribtuion:
µx =
X
µy Pyx =
y
X
y
Cx =
X
y
Cy
Cxy
Cy
; Cx = µx .
Pxy = Cxy /Cx
µy
Cxy
Cy
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Reversible chains and resistors
Thus,
Cxy
Cxy
=:
Pxy = P
.
Cx
z Cxz
Stationary distribtuion:
µx =
X
µy Pyx =
y
X
y
Cx =
X
y
Cy
µy
Cxy
Cy
Cxy
Cy
; Cx = µx .
Pxy = Cxy /Cx
Cx = µx
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Reversible chains and resistors
Thus,
Cxy
Cxy
=:
Pxy = P
.
Cx
z Cxz
Stationary distribtuion:
µx =
X
µy Pyx =
y
X
y
Cx =
X
y
Cy
µy
Cxy
Cy
Cxy
Cy
; Cx = µx .
Notice µx Pxy = Cxy = Cyx = µy Pyx , so the chain is reversible.
Pxy = Cxy /Cx
Cx = µx
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Reversible chains and resistors
Let nx = Ea (number of visits to x before absorbed in b). Then
X
X Cxy
ny
nx =
ny Pyx =
C
y
y
y
Pxy = Cxy /Cx
Cx = µx
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Reversible chains and resistors
Let nx = Ea (number of visits to x before absorbed in b). Then
X
X Cxy
ny
nx =
ny Pyx =
C
y
y
y
ux =
X Cxy
y
Pxy = Cxy /Cx
Cx
· uy
Cx = µx
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Reversible chains and resistors
Let nx = Ea (number of visits to x before absorbed in b). Then
X
X Cxy
ny
nx =
ny Pyx =
C
y
y
y
ux =
X Cxy
y
ux Cx =
X Cxy
y
Pxy = Cxy /Cx
Cx
Cy
· uy
· uy Cy
Cx = µx
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Reversible chains and resistors
Let nx = Ea (number of visits to x before absorbed in b). Then
X
X Cxy
nx =
ny Pyx =
ny
C
y
y
y
ux =
X Cxy
y
ux Cx =
Cx
X Cxy
y
Cy
· uy
· uy Cy
; ux Cx = nx .
Pxy = Cxy /Cx
Cx = µx
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Reversible chains and resistors
Let nx = Ea (number of visits to x before absorbed in b). Then
X
X Cxy
nx =
ny Pyx =
ny
C
y
y
y
ux =
X Cxy
y
ux Cx =
Cx
X Cxy
y
Cy
· uy
· uy Cy
; ux Cx = nx .
Ea (signed current x → y before absorbed in b)
= nx Pxy − ny Pyx = (ux − uy )Cxy = ixy .
Pxy = Cxy /Cx
normalisation. . .
Cx = µx
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Reducing a network
Series:
Q
R
Reff = R + Q
Parallel:
R
Q
1
1
1
= +
Reff
R Q
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Reducing a network
S∗
Star-Delta:
R∆
Q∆
Q∗
R∗
S∆
R∗ =
Q∆ S∆
,
R∆ + Q∆ + S∆
R∆ =
R∗ Q∗ + R∗ S∗ + Q∗ S∗
.
R∗
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Thomson, Dirichlet principles
Thomson principle:
1
R
1
Q
The physical unit current isP
the unit flow that minimizes the sum
of the ohmic power losses
i 2 R.
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Thomson, Dirichlet principles
Thomson principle:
1
R
1
Q
The physical unit current isP
the unit flow that minimizes the sum
of the ohmic power losses
i 2 R.
Dirichlet principle:
1V
R
u
Q
0V
The physical voltage
is the function that minimizes the ohmic
P
power losses (∇u)2 /R.
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Monotonicity, transience, recurrence
The monotonicity property:
Between any disjoint sets of vertices, the effective resistance is
a non-decreasing function of the individual resistances.
Reversible Irreversible Engineering
Reducing a network Thomson, Dirichlet Monotonicity
Monotonicity, transience, recurrence
The monotonicity property:
Between any disjoint sets of vertices, the effective resistance is
a non-decreasing function of the individual resistances.
; can be used to prove transience-recurrence by reducing the
graph to something manageable in terms of resistor networks.
Reversible Irreversible Engineering
The part Chain Network Effective Works...
The part
i
∗λ
x
R/2
R/2
y
Voltage amplifier: keeps the current, multiplies the potential.
(ux − i ·
R
R
) · λ − i · = uy
2
2
Reversible Irreversible Engineering
The part Chain Network Effective Works...
The part
i
∗λ
x
R/2
R/2
y
Voltage amplifier: keeps the current, multiplies the potential.
(ux − i ·
R
R
) · λ − i · = uy
2
2
Reversible Irreversible Engineering
The part Chain Network Effective Works...
The part
i
∗λ
x
R/2
R/2
y
Voltage amplifier: keeps the current, multiplies the potential.
(ux − i ·
R
R
) · λ − i · = uy
2
2
Reversible Irreversible Engineering
The part Chain Network Effective Works...
The part
i
∗λ
x
R/2
R/2
y
Voltage amplifier: keeps the current, multiplies the potential.
(ux − i ·
R
R
) · λ − i · = uy
2
2
Reversible Irreversible Engineering
The part Chain Network Effective Works...
The part
i
∗λ
x
R/2
R/2
y
Voltage amplifier: keeps the current, multiplies the potential.
(ux − i ·
R
R
) · λ − i · = uy
2
2
Reversible Irreversible Engineering
The part Chain Network Effective Works...
The part
i
∗λ
x
R/2
R/2
y
Voltage amplifier: keeps the current, multiplies the potential.
(ux − i ·
R
R
) · λ − i · = uy
2
2
Equivalent:
∗λpr
x
R pr
∗λse
y
(ux − i · R pr ) · λpr = uy
x
R se
y
ux · λse − R se · i = uy
Reversible Irreversible Engineering
The part Chain Network Effective Works...
The part
i
∗λ
x
R/2
R/2
y
Voltage amplifier: keeps the current, multiplies the potential.
(ux − i ·
R
R
) · λ − i · = uy
2
2
Equivalent:
∗λpr
x
R pr
∗λse
y
(ux − i · R pr ) · λpr = uy
λpr = λ
x
R se
y
ux · λse − R se · i = uy
λse = λ
Reversible Irreversible Engineering
The part Chain Network Effective Works...
The part
i
∗λ
x
R/2
R/2
y
Voltage amplifier: keeps the current, multiplies the potential.
(ux − i ·
R
R
) · λ − i · = uy
2
2
Equivalent:
∗λpr
x
R pr
∗λse
y
(ux − i · R pr ) · λpr = uy
λpr = λ
pr
R = λ+1
2λ · R
x
R se
y
ux · λse − R se · i = uy
λse = λ
se
R = λ+1
2 ·R
Reversible Irreversible Engineering
The part Chain Network Effective Works...
The part
i
∗λ
x
R/2
R/2
y
Voltage amplifier: keeps the current, multiplies the potential.
(ux − i ·
R
R
) · λ − i · = uy
2
2
Equivalent:
∗λpr
x
R pr
∗λse
y
(ux − i · R pr ) · λpr = uy
λpr = λ
pr
R = λ+1
2λ · R
R pr =
λ+1
2λ
·R
x
R se
y
ux · λse − R se · i = uy
λse = λ
se
R = λ+1
2 ·R
R se =
λ+1
2
·R
Reversible Irreversible Engineering
The part Chain Network Effective Works...
Harmonicity
∗λxb
∗λxa
a
R pr =
λ+1
2λ
·R
Rxa /2
Rxa /2
x
Rxb /2
Rxb /2
b
R se =
λ+1
2
·R
Reversible Irreversible Engineering
The part Chain Network Effective Works...
Harmonicity
∗λxb
∗λxa
a
Rxa /2
Rxa /2
x
Rxb /2
R pr =
λ+1
2λ
·R
b
∗λxb
∗λxa
a
Rxb /2
se
Rxa
x
se
Rxb
b
R se =
λ+1
2
·R
Reversible Irreversible Engineering
The part Chain Network Effective Works...
Harmonicity
∗λxb
∗λxa
a
Rxa /2
Rxa /2
x
Rxb /2
ux =
X
y
R pr =
λ+1
2λ
·R
b
∗λxb
∗λxa
a
Rxb /2
se
Rxa
x
se
Rxb
b
C se
P xy se · λxy uy
z Cxz
R se =
λ+1
2
·R
Reversible Irreversible Engineering
The part Chain Network Effective Works...
Harmonicity
∗λxb
∗λxa
a
Rxa /2
Rxa /2
x
Rxb /2
ux =
X
y
R pr =
λ+1
2λ
·R
b
∗λxb
∗λxa
a
Rxb /2
se
Rxa
x
se
Rxb
b
C se
P xy se · λxy uy
z Cxz
R se =
λ+1
2
·R
Reversible Irreversible Engineering
The part Chain Network Effective Works...
Harmonicity
∗λxb
∗λxa
a
Rxa /2
Rxa /2
x
Rxb /2
ux =
X
y
R pr =
λ+1
2λ
·R
b
∗λxb
∗λxa
a
Rxb /2
se
Rxa
x
b
se
Rxb
X Cxy λxy2+1
C se
P xy se · λxy uy =
· λxy uy
P
2
z Cxz
z Cxz λ +1
y
xz
R se =
λ+1
2
·R
Reversible Irreversible Engineering
The part Chain Network Effective Works...
Harmonicity
∗λxb
∗λxa
a
Rxa /2
Rxa /2
x
Rxb /2
ux =
X
y
R pr =
λ+1
2λ
·R
b
∗λxb
∗λxa
a
Rxb /2
se
Rxa
x
b
se
Rxb
X Cxy λxy2+1
C se
P xy se · λxy uy =
· λxy uy
P
2
z Cxz
z Cxz λ +1
y
xz
R se =
λ+1
2
·R
Reversible Irreversible Engineering
The part Chain Network Effective Works...
Harmonicity
∗λxb
∗λxa
a
Rxa /2
Rxa /2
x
Rxb /2
ux =
X
=
X
y
y
with γxy =
R pr =
λ+1
2λ
se
Rxa
x
b
se
Rxb
X Cxy λxy2+1
C se
P xy se · λxy uy =
· λxy uy
P
2
z Cxz
z Cxz λ +1
y
xz
D γ
P xy xy · uy
z Dxz γzx
p
λxy =
·R
b
∗λxb
∗λxa
a
Rxb /2
1
γyx ,
Dxy =
2γxy Cxy
(λxy +1)
= Dyx .
R se =
λ+1
2
·R
Reversible Irreversible Engineering
The part Chain Network Effective Works...
Harmonicity
∗λxb
∗λxa
a
Rxa /2
Rxa /2
x
Rxb /2
ux =
X
=
X
y
y
with γxy =
R pr =
λ+1
2λ
se
Rxa
x
b
se
Rxb
X Cxy λxy2+1
C se
P xy se · λxy uy =
· λxy uy
P
2
z Cxz
z Cxz λ +1
y
xz
D γ
P xy xy · uy
z Dxz γzx
p
λxy =
·R
b
∗λxb
∗λxa
a
Rxb /2
1
γyx ,
Dxy =
2γxy Cxy
(λxy +1)
= Dyx .
R se =
λ+1
2
·R
Reversible Irreversible Engineering
The part Chain Network Effective Works...
Harmonicity
∗λxb
∗λxa
a
Rxa /2
Rxa /2
x
Rxb /2
ux =
X
=
X
y
y
with γxy =
R pr =
λ+1
2λ
se
Rxa
x
b
se
Rxb
X Cxy λxy2+1
C se
P xy se · λxy uy =
· λxy uy
P
2
z Cxz
z Cxz λ +1
y
xz
X Dxy γxy
D γ
P xy xy · uy =
· uy
Dx
z Dxz γzx
y
p
λxy =
·R
b
∗λxb
∗λxa
a
Rxb /2
1
γyx ,
Dxy =
2γxy Cxy
(λxy +1)
= Dyx .
R se =
λ+1
2
·R
Reversible Irreversible Engineering
The part Chain Network Effective Works...
Harmonicity
∗λxb
∗λxa
a
Rxa /2
Rxa /2
x
Rxb /2
ux =
X
=
X
y
y
with γxy
γxy =
se
Rxa
λxy
x
b
se
Rxb
X Cxy λxy2+1
C se
P xy se · λxy uy =
· λxy uy
P
2
z Cxz
z Cxz λ +1
y
xz
X Dxy γxy
D γ
P xy xy · uy =
· uy
Dx
z Dxz γzx
y
p
= λxy =
p
b
∗λxb
∗λxa
a
Rxb /2
1
γyx ,
Dx =
Dxy =
X
z
Dxz γzx
2γxy Cxy
(λxy +1)
= Dyx .
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
From network to chain
Irreducible Markov chain: on Ω, a 6= b, x ∈ Ω,
hx : = Px {τa < τb }
(τ is the hitting time)
is harmonic:
hx =
X
Pxy hy ,
ha = 1,
hb = 0.
y
ux =
X Dxy γxy
y
γxy =
p
λxy
Dx
Dx =
X
z
· uy ,
Dxz γzx
ua = 1,
ub = 0.
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
From network to chain
Irreducible Markov chain: on Ω, a 6= b, x ∈ Ω,
hx : = Px {τa < τb }
(τ is the hitting time)
is harmonic:
hx =
X
Pxy hy ,
ha = 1,
hb = 0.
y
ux =
X Dxy γxy
y
Dx
· uy ,
Pxy =
γxy =
p
λxy
Dx =
X
z
ua = 1,
ub = 0.
Dxy γxy
.
Dx
Dxz γzx
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
From network to chain
Irreducible Markov chain: on Ω, a 6= b, x ∈ Ω,
hx : = Px {τa < τb }
(τ is the hitting time)
is harmonic:
hx =
X
Pxy hy ,
ha = 1,
hb = 0.
y
ux =
X Dxy γxy
y
Dx
· uy ,
Pxy =
γxy =
p
λxy
Dx =
X
ua = 1,
ub = 0.
Dxy γxy
.
Dx
Dxz γzx
z
Pxy = Dxy γxy /Dx
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
From network to chain
Stationary distribtuion:
µx =
X
µz Pzx =
z
z
Dx =
X
Dz
z
γxy =
p
λxy
Dx =
X
X
Dxz γzx
z
Pxy = Dxy γxy /Dx
µz
Dzx γzx
Dz
Dzx γzx
Dz
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
From network to chain
Stationary distribtuion:
µx =
X
µz Pzx =
z
X
µz
z
Dx =
X
Dz
z
Dzx γzx
Dz
Dzx γzx
Dz
; Dx = µx .
γxy =
p
λxy
Dx =
X
Dxz γzx
z
Pxy = Dxy γxy /Dx
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
”Markovian” property
ux =
X
Pxz uz ;
z
X
Pxz = 1
z
ux ≡ const. is a solution of the network with no external
sources. This is now nontrivial.
γxy =
p
λxy
Dx =
X
Dxz γzx
z
Pxy = Dxy γxy /Dx
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
”Markovian” property
ux =
X
Pxz uz ;
z
X
Pxz = 1
z
ux ≡ const. is a solution of the network with no external
sources. This is now nontrivial.
X
Pxz =
z
X Dxz γxz
z
Dx : =
X
Dx
=1
Dxz γzx
z
γxy =
p
λxy
Dx =
X
Dxz γzx
z
Pxy = Dxy γxy /Dx
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
”Markovian” property
ux =
X
X
Pxz uz ;
z
Pxz = 1
z
ux ≡ const. is a solution of the network with no external
sources. This is now nontrivial.
X
Pxz =
z
z
Dx : =
X
p
λxy
Dx =
X
Dx
Dxz γzx =
z
γxy =
X Dxz γxz
Dxz γzx
z
Pxy = Dxy γxy /Dx
X
=1
Dxz γxz .
z
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
”Markovian” property
ux =
X
X
Pxz uz ;
z
Pxz = 1
z
ux ≡ const. is a solution of the network with no external
sources. This is now nontrivial.
X
Pxz =
z
X Dxz γxz
z
Dx : =
X
Dxz γzx =
z
γxy =
p
λxy
Dx =
X
Dx
Dxz γzx =
z
Pxy = Dxy γxy /Dx
X
=1
Dxz γxz .
z
X
z
Dxz γxz
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
From chain to network
Pxy =
Dxy γxy
Dxy γxy
=
Dx
µx
2
µx Pxy · µy Pyx = Dxy
;
µx Pxy
2
= γxy
= λxy .
µy Pyx
γxy =
p
λxy
Dx =
X
Dxz γzx =
z
Pxy = Dxy γxy /Dx
X
z
Dxz γxz
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
From chain to network
Pxy =
Dxy γxy
Dxy γxy
=
Dx
µx
2
µx Pxy · µy Pyx = Dxy
;
µx Pxy
2
= γxy
= λxy .
µy Pyx
Reversed chain: Replace Pxy by P̂xy = Pyx ·
µy
µx .
; Dxy stays, λxy reverses to λyx .
γxy =
p
λxy
Dx =
X
Dxz γzx =
z
Pxy = Dxy γxy /Dx
X
z
Dxz γxz
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
From chain to network
Let nx = Ea (number of visits to x before absorbed in b). Then
nx =
X
ny Pyx =
y
γxy =
p
λxy
Dx =
X
Dxz γzx =
z
Pxy = Dxy γxy /Dx
X Dyx γyx
y
X
z
Dxz γxz
Dy
ny
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
From chain to network
Let nx = Ea (number of visits to x before absorbed in b). Then
nx =
X
ny Pyx =
y
ux =
p
λxy
Dx =
X
Dy
y
X Dxy γxy
Dx
y
γxy =
X Dyx γyx
Dxz γzx =
z
Pxy = Dxy γxy /Dx
X
z
ny
· uy
Dxz γxz
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
From chain to network
Let nx = Ea (number of visits to x before absorbed in b). Then
nx =
X
ny Pyx =
y
ux =
X Dxy γxy
Dx
X Dxy γxy
Dy
y
γxy =
p
λxy
Dx =
X
Dy
y
y
ux Dx =
X Dyx γyx
Dxz γzx =
z
Pxy = Dxy γxy /Dx
X
z
ny
· uy
· uy Dy
Dxz γxz
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
From chain to network
Let nx = Ea (number of visits to x before absorbed in b). Then
nx =
X
ny Pyx =
X Dyx γyx
Dy
y
y
ux =
X Dxy γxy
Dx
y
ux Dx =
X Dxy γxy
Dy
y
ny
· uy
· uy Dy
; ûx Dx = nx
in the reversed chain.
γxy =
p
λxy
Dx =
X
Dxz γzx =
z
Pxy = Dxy γxy /Dx
X
z
Dxz γxz
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
From chain to network
Let nx = Ea (number of visits to x before absorbed in b). Then
; ûx Dx = nx
in the reversed chain.
Ea (signed current x → y before absorbed in b)
= nx Pxy −ny Pyx = (ûx γxy − ûy γyx )Dxy = îxy .
γxy =
p
λxy
Dx =
X
Dxz γzx =
z
Pxy = Dxy γxy /Dx
X
z
Dxz γxz
normalisation. . .
Dxy = 2γxy Cxy /(λxy + 1)
Reversible Irreversible Engineering
The part Chain Network Effective Works...
Effective resistance
Suppose ua , ub given, the solution is {ux }x∈Ω and {ixy }x∼y ∈Ω .
Current
X
iax
ia =
x∼a
flows in the network at a.
Reversible Irreversible Engineering
The part Chain Network Effective Works...
Effective resistance
Suppose ua , ub given, the solution is {ux }x∈Ω and {ixy }x∼y ∈Ω .
Current
X
iax
ia =
x∼a
flows in the network at a.
;The ”Markovian” property has another solution: constant ub
potentials with zero external currents.
Reversible Irreversible Engineering
The part Chain Network Effective Works...
Effective resistance
Suppose ua , ub given, the solution is {ux }x∈Ω and {ixy }x∼y ∈Ω .
Current
X
iax
ia =
x∼a
flows in the network at a.
;The ”Markovian” property has another solution: constant ub
potentials with zero external currents.
; The difference of these two: {ux − ub }x∈Ω is a solution too,
with ia flowing in the network.
Reversible Irreversible Engineering
The part Chain Network Effective Works...
Effective resistance
Suppose ua , ub given, the solution is {ux }x∈Ω and {ixy }x∼y ∈Ω .
Current
X
iax
ia =
x∼a
flows in the network at a.
;The ”Markovian” property has another solution: constant ub
potentials with zero external currents.
; The difference of these two: {ux − ub }x∈Ω is a solution too,
with ia flowing in the network.
; Going backwards from ub − ub = 0 at b, all currents and
potentials are proportional to ua − ub at a.
Reversible Irreversible Engineering
The part Chain Network Effective Works...
Effective resistance
Suppose ua , ub given, the solution is {ux }x∈Ω and {ixy }x∼y ∈Ω .
Current
X
iax
ia =
x∼a
flows in the network at a.
;The ”Markovian” property has another solution: constant ub
potentials with zero external currents.
; The difference of these two: {ux − ub }x∈Ω is a solution too,
with ia flowing in the network.
; Going backwards from ub − ub = 0 at b, all currents and
potentials are proportional to ua − ub at a.
; In particular, ia is proportional to ua − ub . We have effective
resistance.
Reversible Irreversible Engineering
What works
... the analogy with P{τa < τb }.
The part Chain Network Effective Works...
Reversible Irreversible Engineering
The part Chain Network Effective Works...
What works
... the analogy with P{τa < τb }.
Modulo normalisation. . .
Ea (signed current x → y before absorbed in b) = îxy .
in the reversed network!
Reversible Irreversible Engineering
The part Chain Network Effective Works...
What works
... the analogy with P{τa < τb }.
Modulo normalisation. . .
Ea (signed current x → y before absorbed in b) = îxy .
in the reversed network!
Theorem (Chandra, Raghavan, Ruzzo, Smolensky and
Tiwari ’96 for reversible)
Commute time = Reff · all conductances.
Reversible Irreversible Engineering
The part Chain Network Effective Works...
What works
For all sets A, B, capacity∼escape probability.
eff
cap(A, B) = CAB
=
1
eff
RAB
Reversible Irreversible Engineering
The part Chain Network Effective Works...
What works
For all sets A, B, capacity∼escape probability.
eff
cap(A, B) = CAB
=
This is non-physical!
1 X
1
=
Cxy (ux − uy )2 .
eff
2
RAB
x∼y ∈V
Reversible Irreversible Engineering
The part Chain Network Effective Works...
What works
For all sets A, B, capacity∼escape probability.
eff
cap(A, B) = CAB
=
1 X
1
=
Cxy (ux − uy )2 .
eff
2
RAB
x∼y ∈V
This is non-physical!
In particular, symmetrising the chain (Pxy →
increase escape probabilities:
Pxy +P̂xy
)
2
cannot
◮
symmetrising leaves Cxy unchanged;
◮
the above sum is minimised by the symmetric voltages, not
{ux } (Classical Dirichlet principle).
Reversible Irreversible Engineering
Reducing Nonmonotonicity Dirichlet
The electric network
Series:
∗µ
∗λ
R/2
R/2
Q/2
∗µ
∗λ
R
Q/2
Q se
pr
∗λµ
Q se
∗λµ
R pr
R pr
Q′
∗λµ
pr
S pr
∗λµ
S/2
S=R
S/2
µ+1
(λ + 1)µ
+Q
.
λµ + 1
λµ + 1
Reversible Irreversible Engineering
Reducing Nonmonotonicity Dirichlet
The electric network
Parallel:
∗λ
R/2
∗λ
R se
R/2
∗µ
Q/2
∗µ
Q se
Q/2
Compare this with
∗ν
S se
RQ
R+Q
Qλ(µ + 1) + Rµ(λ + 1)
ν=
.
Q(µ + 1) + R(λ + 1)
S=
Reversible Irreversible Engineering
Reducing Nonmonotonicity Dirichlet
The electric network
Star-Delta:
Star to Delta works,
Delta to Star only works if Delta does not produce a circular
current by itself (λµν = 1).
R/
2
2
Q/
2
Q/
R/
2
∗λ
∗µ
∗ν
S/2
S/2
Reversible Irreversible Engineering
Reducing Nonmonotonicity Dirichlet
Nonmonotonicity
∗1/5
3/2
∗5/13
3/2
R
∗5
3/2
R eff =
2/2
3/2
2/2
∗13/5
2/2
1296
27
+
.
14 1225R + 2268
2/2
Reversible Irreversible Engineering
Reducing Nonmonotonicity Dirichlet
Nonmonotonicity
∗1/5
3/2
∗5/13
3/2
R
∗5
3/2
R eff =
2/2
3/2
2/2
∗13/5
2/2
1296
27
+
.
14 1225R + 2268
2/2
/
Reversible Irreversible Engineering
Dirichlet principle
Classical case:
Reducing Nonmonotonicity Dirichlet
Reversible Irreversible Engineering
Reducing Nonmonotonicity Dirichlet
Dirichlet principle
Classical case:
(iu )xy = Cxy · u(x) − u(y) ,
X
EOhm (iu )=
(iu )2xy · Rxy .
x∼y
Reversible Irreversible Engineering
Reducing Nonmonotonicity Dirichlet
Dirichlet principle
Classical case:
eff
Rab
=
EOhm (iu ),
(iu )xy = Cxy · u(x) − u(y) ,
X
EOhm (iu )=
(iu )2xy · Rxy .
x∼y
Reversible Irreversible Engineering
Reducing Nonmonotonicity Dirichlet
Dirichlet principle
Classical case:
eff
Rab
=
min
u:u(a)=1, u(b)=0
EOhm (iu ),
(iu )xy = Cxy · u(x) − u(y) ,
X
EOhm (iu )=
(iu )2xy · Rxy .
x∼y
Reversible Irreversible Engineering
Reducing Nonmonotonicity Dirichlet
Dirichlet principle
Classical case:
eff
Rab
=
min
u:u(a)=1, u(b)=0
EOhm (iu ),
(iu )xy = Cxy · u(x) − u(y) ,
X
EOhm (iu )=
(iu )2xy · Rxy .
x∼y
Irreversible case (A. Gaudillière, C. Landim / M. Slowik):
(iu∗ )xy = Dxy · γxy u(x) − γyx u(y) ,
X
2
iu∗ − Ψxy · Rxy .
EOhm (iu∗ − Ψ)=
x∼y
Reversible Irreversible Engineering
Reducing Nonmonotonicity Dirichlet
Dirichlet principle
Classical case:
eff
Rab
=
min
u:u(a)=1, u(b)=0
EOhm (iu ),
(iu )xy = Cxy · u(x) − u(y) ,
X
EOhm (iu )=
(iu )2xy · Rxy .
x∼y
Irreversible case (A. Gaudillière, C. Landim / M. Slowik):
eff
Rab
=
EOhm (iu∗ − Ψ),
(iu∗ )xy = Dxy · γxy u(x) − γyx u(y) ,
X
2
iu∗ − Ψxy · Rxy .
EOhm (iu∗ − Ψ)=
x∼y
Reversible Irreversible Engineering
Reducing Nonmonotonicity Dirichlet
Dirichlet principle
Classical case:
eff
Rab
=
min
u:u(a)=1, u(b)=0
EOhm (iu ),
(iu )xy = Cxy · u(x) − u(y) ,
X
EOhm (iu )=
(iu )2xy · Rxy .
x∼y
Irreversible case (A. Gaudillière, C. Landim / M. Slowik):
eff
Rab
=
min EOhm (iu∗ − Ψ),
Ψ: flow
(iu∗ )xy = Dxy · γxy u(x) − γyx u(y) ,
X
2
iu∗ − Ψxy · Rxy .
EOhm (iu∗ − Ψ)=
x∼y
Reversible Irreversible Engineering
Reducing Nonmonotonicity Dirichlet
Dirichlet principle
Classical case:
eff
Rab
=
min
u:u(a)=1, u(b)=0
EOhm (iu ),
(iu )xy = Cxy · u(x) − u(y) ,
X
EOhm (iu )=
(iu )2xy · Rxy .
x∼y
Irreversible case (A. Gaudillière, C. Landim / M. Slowik):
eff
Rab
=
min EOhm (iu∗ − Ψ),
min
u:u(a)=1, u(b)=0 Ψ: flow
(iu∗ )xy = Dxy · γxy u(x) − γyx u(y) ,
X
2
iu∗ − Ψxy · Rxy .
EOhm (iu∗ − Ψ)=
x∼y
Reversible Irreversible Engineering
Reducing Nonmonotonicity Dirichlet
Dirichlet principle
Classical case:
eff
Rab
=
min
u:u(a)=1, u(b)=0
EOhm (iu ),
(iu )xy = Cxy · u(x) − u(y) ,
X
EOhm (iu )=
(iu )2xy · Rxy .
x∼y
Irreversible case (A. Gaudillière, C. Landim / M. Slowik):
eff
Rab
=
min EOhm (iu∗ − Ψ),
min
u:u(a)=1, u(b)=0 Ψ: flow
(iu∗ )xy = Dxy · γxy u(x) − γyx u(y) ,
X
2
iu∗ − Ψxy · Rxy .
EOhm (iu∗ − Ψ)=
x∼y
Thank you.
Reversible Irreversible Engineering
A trivial statement
Theorem (Well Known Theorem)
A Markov chain is reversible if and only if for every closed cycle
x0 , x1 , x2 , . . . , xn = x0 in Ω we have
Px0 x1 · Px1 x2 · · · Pxn−1 x0 = Px0 xn−1 · Pxn−1 xn−2 · · · Px1 x0 .
In particular, any Markov chain on a finite connected tree G is
necessarily reversible.
Reversible Irreversible Engineering
A trivial statement
Electrical proof.
Plug in
Pxy =
Dxy γxy
,
Dx
Dxy symmetric:
Px0 x1 · Px1 x2 · · · Pxn−1 x0 = Px0 xn−1 · Pxn−1 xn−2 · · · Px1 x0
γx0 x1 · γx1 x2 · · · γxn−1 x0 = γx0 xn−1 · γxn−1 xn−2 · · · γx1 x0
λx0 x1 · λx1 x2 · · · λxn−1 x0 = 1.
◮
Total multiplication factor along any loop is one.
, or
Reversible Irreversible Engineering
A trivial statement
Electrical proof.
Plug in
Pxy =
Dxy γxy
,
Dx
Dxy symmetric:
Px0 x1 · Px1 x2 · · · Pxn−1 x0 = Px0 xn−1 · Pxn−1 xn−2 · · · Px1 x0
γx0 x1 · γx1 x2 · · · γxn−1 x0 = γx0 xn−1 · γxn−1 xn−2 · · · γx1 x0
λx0 x1 · λx1 x2 · · · λxn−1 x0 = 1.
◮
Total multiplication factor along any loop is one.
◮
Zero current and free vertices is a solution.
, or
Reversible Irreversible Engineering
A trivial statement
Electrical proof.
Plug in
Pxy =
Dxy γxy
,
Dx
Dxy symmetric:
Px0 x1 · Px1 x2 · · · Pxn−1 x0 = Px0 xn−1 · Pxn−1 xn−2 · · · Px1 x0
γx0 x1 · γx1 x2 · · · γxn−1 x0 = γx0 xn−1 · γxn−1 xn−2 · · · γx1 x0
λx0 x1 · λx1 x2 · · · λxn−1 x0 = 1.
◮
Total multiplication factor along any loop is one.
◮
Zero current and free vertices is a solution.
◮
It’s the only solution.
, or
Reversible Irreversible Engineering
A trivial statement
Electrical proof.
Plug in
Pxy =
Dxy γxy
,
Dx
Dxy symmetric:
Px0 x1 · Px1 x2 · · · Pxn−1 x0 = Px0 xn−1 · Pxn−1 xn−2 · · · Px1 x0
γx0 x1 · γx1 x2 · · · γxn−1 x0 = γx0 xn−1 · γxn−1 xn−2 · · · γx1 x0
λx0 x1 · λx1 x2 · · · λxn−1 x0 = 1.
◮
Total multiplication factor along any loop is one.
◮
Zero current and free vertices is a solution.
◮
It’s the only solution.
◮
The network is ”Markovian”: potential is constant.
, or
Reversible Irreversible Engineering
A trivial statement
Electrical proof.
Plug in
Pxy =
Dxy γxy
,
Dx
Dxy symmetric:
Px0 x1 · Px1 x2 · · · Pxn−1 x0 = Px0 xn−1 · Pxn−1 xn−2 · · · Px1 x0
γx0 x1 · γx1 x2 · · · γxn−1 x0 = γx0 xn−1 · γxn−1 xn−2 · · · γx1 x0
λx0 x1 · λx1 x2 · · · λxn−1 x0 = 1.
◮
Total multiplication factor along any loop is one.
◮
Zero current and free vertices is a solution.
◮
It’s the only solution.
◮
The network is ”Markovian”: potential is constant.
◮
All λ’s are 1, and the chain is reversible.
, or
Reversible Irreversible Engineering
A trivial statement
Electrical proof.
Repeat for trees:
◮
There are no loops.
Reversible Irreversible Engineering
A trivial statement
Electrical proof.
Repeat for trees:
◮
There are no loops.
◮
Zero current and free vertices is a solution.
Reversible Irreversible Engineering
A trivial statement
Electrical proof.
Repeat for trees:
◮
There are no loops.
◮
Zero current and free vertices is a solution.
◮
It’s the only solution.
Reversible Irreversible Engineering
A trivial statement
Electrical proof.
Repeat for trees:
◮
There are no loops.
◮
Zero current and free vertices is a solution.
◮
It’s the only solution.
◮
The network is ”Markovian”: potential is constant.
Reversible Irreversible Engineering
A trivial statement
Electrical proof.
Repeat for trees:
◮
There are no loops.
◮
Zero current and free vertices is a solution.
◮
It’s the only solution.
◮
The network is ”Markovian”: potential is constant.
◮
All λ’s are 1, and the chain is reversible.
Reversible Irreversible Engineering
A trivial statement
Electrical proof.
Repeat for trees:
◮
There are no loops.
◮
Zero current and free vertices is a solution.
◮
It’s the only solution.
◮
The network is ”Markovian”: potential is constant.
◮
All λ’s are 1, and the chain is reversible.
Second thank you.