Our interest is in real symmetric conservative matrices.
\[ \mathbf{W} = \left(\matrix{
-\gamma_{1} & w_{12} & w_{13} & \cdots & \cdots \cr
w_{12} & -\gamma_2 & w_{23} & \cr
w_{13} & w_{23} & -\gamma_3 & \cr
\vdots & & & \ddots\cr
\vdots & & & & \ddots}\right) \]
The "conservative" means that each row's sum is zero, or:
\[\gamma_n \ \ \equiv \ \ \sum_{m\neq n} w_{mn} \]
Motivation
Such matrices appear in rate equations:
\[ \frac{d}{dt}{\mathbf p} \ \ = \ \ \mathbf{W}\cdot\mathbf{p} \]
in Newtons II law for systems of masses and springs, which are used as heat transport models:
\[ \frac{d^2}{dt^2}{\mathbf q} \ \ = \ \ -\mathbf{K}\cdot\mathbf{q} \]
and in Schrödinger's equation
\[ \frac{d}{dt}{\mathbf \psi} \ \ = \ \ -i {\mathcal{H}}\cdot\mathbf{\psi} \]
Sparsity
We focus on sparse/glassy networks,
where by sparse we mean that a small number of elements
is larger by orders of magnitude than the rest.
As two examples we use:
Response of driven weakly chaotic systems
Mott random site model
In this model, the sites are
randomly distributed in space, and the rates depend on
the distance between sites:
\[ w_{nm} \ \ =\ \ w_0 \cdot \exp\left(-\frac{\left|r_n-r_m\right|}{\xi} - \epsilon_{nm}\right)\]
Where \(\xi\) and \(w_0\) are system parameters (defining the time and space units),
and \(\epsilon\) a bond specific parameter. In the degenerate version, \(\epsilon=0\)
for all the bonds, while in the non-degenerate version \(\epsilon_{nm} = \textrm{uniform}[0,\infty]\).
The dimensionless parameter defining sparsity in this model is
\[ s \ \ =\ \ \frac{\xi}{r_0} \]
where \(r_0\) is the typical distance between sites.
Determining the diffusion coefficient \(D\)
The long term behavior of the system is characterized by
a diffusion coefficient \(D\). It is defined by the long term spreading:
\[ S(t)\quad = \quad \left\langle r^2(t)\right\rangle \quad \sim \quad D t\]
And it is also related to the long time survival probability:
\[ \mathcal{P}(t)\quad \sim \quad \frac{1}{(Dt)^{1/2}} \]
And to the low eigenvalue distribution (by Laplace transform):
\[ \mathcal{N}(\lambda) \quad \sim \quad \left[ \frac{\lambda}{D}\right]^{d/2} \]
Effective range hopping
For a dense system, (large \(\xi\) compared to typical site distance),
the diffusion coefficient can be estimated by a linear equation (linear in the rates):
\[ D_{\textrm{linear}} = \frac{1}{2d}\sum_r w(r) r^2 \]
We present the ERH procedure to estimate \(D\),
based on resistor network analysis, with a smooth cross-over between the
dense and sparse regimes.
The parameter \(n_c\) is the average number of bonds required
to get percolation. For \(d=1\) \(n_c=2\) and for \(d=2\) \(n_c\approx 4.5\),
based on studies of disk-percolation.
If we disregard the percolative nature (by setting \(n_c=0\)), we
get the linear estimate back.
In the limit \(s_\textrm{eff}\ll 1\) we obtain VRH-like behavior:
\[ D\sim e^{-1/s_\textrm{eff}} \]
Results for the non-degenerate "Mott" model
In this case, \(s_\textrm{eff}\) depends explicitly
on the temperature, i.e.:
\[s_{\textrm{eff}}\quad=\quad \left(\frac{d}{\Omega_d} n_c \left(\frac{T}{\Delta_\xi}\right)\right)^{-1/(d+1)} \]
Where \(\Delta_\xi\) is the mean level spacing.
Apart from that, the only change in \(D\) is the polynomial order:
\[ D \quad =\quad \textrm{EXP}_{\color{red}{d+3}}\left(\frac{1}{s_{\textrm{eff}}}\right) \mbox{e}^{-1/s_{\textrm{eff}}} D_{linear} \]
In the limit \(s\ll 1\), we get the familiar VRH estimate that has been
presented long ago by Nevill Francis Mott.
\[ D \sim \left(\frac{1}{T}\right)^{2/(d+1)} \exp\left[-\left(\frac{T_0}{T}\right)^{1/(d+1)}\right]\]
Diffusion in sparse networks: linear to semi-linear crossover
[arXiv]
[pdf],
Y. de Leeuw, D. Cohen, Phys. Rev. E 86, 051120 (2012).