藏本模型 （Kuramoto model）是一种用来描述同步 的数学模型 ，由日本物理学家藏本由纪 （Kuramoto Yoshiki）首先提出[1] [2] 振子 的同步行为[3] [4] 神经振荡 [5] [6] [7] [8] [9] 约瑟夫森结 的阵列[10]
这个模型假设，所有振子都是完全相同的或几乎完全相同的，相互之间的耦合很弱、并且任意两个振子之间的相互作用强度取决于它们相位差的正弦。
在藏本模型最常见的版本中，每个振子都有一个固有的自然频率
ω
i
{\displaystyle \omega _{i}}
N
→
∞
{\displaystyle N\to \infty }
非线性 的模型是可以精确求解的。
藏本模型中的锁相 这个模型最常见的形式由以下方程组给出：
d
θ
i
d
t
=
ω
i
+
K
N
∑
j
=
1
N
sin
(
θ
j
−
θ
i
)
,
i
=
1
,
⋯
,
N
{\displaystyle {\frac {d\theta _{i}}{dt}}=\omega _{i}+{\frac {K}{N}}\sum _{j=1}^{N}{\sin {(\theta _{j}-\theta _{i})}},\quad i=1,\cdots ,N}
系统由
N
{\displaystyle N}
θ
i
{\displaystyle \theta _{i}}
i
{\displaystyle i}
K
{\displaystyle K}
也可以在系统中加入噪声。这种情况下，方程变为
d
θ
i
d
t
=
ω
i
+
K
N
∑
j
=
1
N
sin
(
θ
j
−
θ
i
)
+
ζ
i
,
i
=
1
,
⋯
,
N
{\displaystyle {\frac {d\theta _{i}}{dt}}=\omega _{i}+{\frac {K}{N}}\sum _{j=1}^{N}{\sin {(\theta _{j}-\theta _{i})}}+\zeta _{i},\quad i=1,\cdots ,N}
其中
ζ
i
{\displaystyle \zeta _{i}}
⟨
ζ
i
(
t
)
⟩
=
0
,
{\displaystyle \langle \zeta _{i}(t)\rangle =0,}
⟨
ζ
i
(
t
)
ζ
j
(
t
′
)
⟩
=
2
D
δ
i
j
δ
(
t
−
t
′
)
{\displaystyle \langle \zeta _{i}(t)\zeta _{j}(t')\rangle =2D\delta _{ij}\delta (t-t')}
其中
D
{\displaystyle D}
使得这个模型（至少在
N
→
∞
{\displaystyle N\to \infty }
定义“序”参量
R
e
i
ψ
=
1
N
∑
j
=
1
N
e
i
θ
j
{\displaystyle Re^{i\psi }={\frac {1}{N}}\sum _{j=1}^{N}{e^{i\theta _{j}}}}
R
{\displaystyle R}
相关性 ，
ψ
{\displaystyle \psi }
e
−
i
θ
i
{\displaystyle e^{-{\text{i}}\theta _{i}}}
d
θ
i
d
t
=
ω
i
+
K
R
sin
(
ψ
−
θ
i
)
{\displaystyle {\frac {d\theta _{i}}{dt}}=\omega _{i}+KR\sin {(\psi -\theta _{i})}}
因此振子的方程组就不是显式耦合的；相反，序参量支配了系统的行为。通常还会做进一步的变换，变换到一个转动的坐标系，其中所有振子相位的统计平均为零（即
ψ
=
0
{\displaystyle \psi =0}
d
θ
i
d
t
=
ω
i
−
K
R
sin
θ
i
{\displaystyle {\frac {d\theta _{i}}{dt}}=\omega _{i}-KR\sin {\theta _{i}}}
考虑
N
→
∞
{\displaystyle N\to \infty }
g
(
ω
)
{\displaystyle g(\omega )}
t
{\displaystyle t}
ω
{\displaystyle \omega }
θ
{\displaystyle \theta }
ρ
(
θ
,
ω
,
t
)
{\displaystyle \rho (\theta ,\omega ,t)}
∫
0
2
π
ρ
(
θ
,
ω
,
t
)
d
θ
=
1
{\displaystyle \int _{0}^{2\pi }{\rho (\theta ,\omega ,t)d\theta }=1}
振子密度的连续性方程 为
∂
ρ
∂
t
+
∂
(
ρ
v
)
∂
θ
=
0
{\displaystyle {\frac {\partial \rho }{\partial t}}+{\frac {\partial (\rho v)}{\partial \theta }}=0}
其中
v
=
ω
+
K
R
sin
(
ψ
−
θ
)
{\displaystyle v=\omega +KR\sin {(\psi -\theta )}}
最终，在连续统极限下重新写出序参量。
θ
i
{\displaystyle \theta _{i}}
R
e
i
ψ
=
∫
0
2
π
∫
−
∞
∞
ρ
(
θ
,
ω
,
t
)
g
(
ω
)
e
i
θ
d
ω
d
θ
{\displaystyle Re^{i\psi }=\int _{0}^{2\pi }{\int _{-\infty }^{\infty }{\rho (\theta ,\omega ,t)g(\omega )e^{i\theta }d\omega }d\theta }}
所有振子随机漂移的不相关 态对应均匀分布解
ρ
=
1
2
π
{\displaystyle \rho ={\frac {1}{2\pi }}}
R
=
0
{\displaystyle R=0}
稳定态 ，尽管每个振子单独来看都在以自然频率不停运动。
当耦合足够强时，可能会出现完全同步的解。在完全同步态中，所有振子以相同频率运动，但相位可以不同。
部分同步是只有一些振子同步，而另一些振子自由漂移的状态。从数学上来说，对锁相的振子
ρ
=
δ
(
θ
−
ψ
−
arcsin
ω
K
R
)
{\displaystyle \rho =\delta \left(\theta -\psi -\arcsin {\frac {\omega }{KR}}\right)}
对漂移的振子，
ρ
∝
1
ω
−
K
R
sin
(
θ
−
ψ
)
{\displaystyle \rho \propto {\frac {1}{\omega -KR\sin {(\theta -\psi )}}}}
耗散的藏本模型包含在某些保守的哈密顿系统 中[11] 哈密顿量 具有形式：
H
=
∑
i
=
1
N
1
2
ω
i
(
q
i
2
+
p
i
2
)
+
K
4
N
∑
i
,
j
=
1
N
(
q
i
p
j
−
q
j
p
i
)
(
q
j
2
+
p
j
2
−
q
i
2
−
p
i
2
)
{\displaystyle {\mathcal {H}}=\sum _{i=1}^{N}{{\frac {1}{2}}\omega _{i}(q_{i}^{2}+p_{i}^{2})}+{\frac {K}{4N}}\sum _{i,j=1}^{N}{(q_{i}p_{j}-q_{j}p_{i})(q_{j}^{2}+p_{j}^{2}-q_{i}^{2}-p_{i}^{2})}}
用正则变换变成作用量-角度的形式，作用量为
I
i
=
1
2
(
q
i
2
+
p
i
2
)
{\displaystyle I_{i}={\frac {1}{2}}(q_{i}^{2}+p_{i}^{2})}
θ
i
=
arctan
q
i
p
i
{\displaystyle \theta _{i}=\arctan {\frac {q_{i}}{p_{i}}}}
I
i
≡
I
{\displaystyle I_{i}\equiv I}
H
=
∑
i
=
1
N
ω
i
I
i
−
K
N
∑
i
=
1
N
∑
j
=
1
N
I
j
I
i
(
I
j
−
I
i
)
sin
(
θ
j
−
θ
i
)
{\displaystyle {\mathcal {H}}=\sum _{i=1}^{N}{\omega _{i}I_{i}}-{\frac {K}{N}}\sum _{i=1}^{N}{\sum _{j=1}^{N}{{\sqrt {I_{j}I_{i}}}(I_{j}-I_{i})\sin {(\theta _{j}-\theta _{i})}}}}
哈密顿运动方程为
d
I
i
d
t
=
−
∂
H
∂
θ
i
=
−
2
K
N
∑
j
=
1
N
I
j
I
i
(
I
j
−
I
i
)
cos
(
θ
j
−
θ
i
)
{\displaystyle {\frac {dI_{i}}{dt}}=-{\frac {\partial {\mathcal {H}}}{\partial \theta _{i}}}=-{\frac {2K}{N}}\sum _{j=1}^{N}{{\sqrt {I_{j}I_{i}}}(I_{j}-I_{i})\cos {(\theta _{j}-\theta _{i})}}}
d
θ
i
d
t
=
∂
H
∂
I
i
=
ω
i
+
K
N
∑
j
=
1
N
I
j
/
I
i
(
I
i
+
I
j
)
sin
(
θ
j
−
θ
i
)
{\displaystyle {\frac {d\theta _{i}}{dt}}={\frac {\partial {\mathcal {H}}}{\partial I_{i}}}=\omega _{i}+{\frac {K}{N}}\sum _{j=1}^{N}{{\sqrt {I_{j}/I_{i}}}(I_{i}+I_{j})\sin {(\theta _{j}-\theta _{i})}}}
因为
d
I
i
d
t
=
0
{\displaystyle {\frac {dI_{i}}{dt}}=0}
I
i
=
I
{\displaystyle I_{i}=I}
d
θ
i
d
t
{\displaystyle {\frac {d\theta _{i}}{dt}}}
玻色-爱因斯坦凝聚 。
模型有两种类型的变体，一种改变模型的拓扑结构，另一种改变耦合函数的形式。
除了具有全连拓扑的原始模型，足够稠密的复杂网络 拓扑也可以用同样的平均场处理[12]
藏本把两个振子之间的相位相互作用用第1个傅里叶分量来近似，即
Γ
(
ϕ
)
=
sin
ϕ
{\displaystyle \Gamma (\phi )=\sin \phi }
ϕ
=
θ
j
−
θ
i
{\displaystyle \phi =\theta _{j}-\theta _{i}}
Γ
(
ϕ
)
=
sin
ϕ
+
a
1
sin
(
2
ϕ
+
b
1
)
+
⋯
+
a
n
sin
(
2
n
ϕ
+
b
n
)
{\displaystyle \Gamma (\phi )=\sin \phi +a_{1}\sin {(2\phi +b_{1})}+\cdots +a_{n}\sin {(2n\phi +b_{n})}}
例如，对于弱耦合Hodgkin-Huxley神经元的网络，其同步行为可以用一些振子来表示，这些振子的相互作用函数保留前四阶傅里叶分量[13] 异宿环 [14] [15] [16]
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