# 复杂性和网络 Complexity & Networks PHYS96008

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Additionally all path-specific departure rates are non-negative so we write
$$h=\left(h_{\mathrm{p}}: p \in P\right) \geq 0$$
where $P$ is the set of all network paths. As a consequence
$$\Psi_{n}\left(t, h^{}\right)>v_{i i} p \in P_{i i} \Rightarrow h_{n}^{}=0 .$$

as can easily be proven from by contradiction. We next comment that the relevant notion of flow conservation is
$$\sum_{p \in P_{j}} \int_{0}^{T} h_{\mathrm{p}}(t) \mathrm{d} t=Q_{i j} \quad \forall(i, j) \in W$$
where $Q_{i j}$ is the fixed travel demand (expressed as a traffic volume) for $(i, j) \in W$. Thus, the set of feasible solutions is

## PHYS96008 COURSE NOTES ：

We can then seek to work with the free energy and the model Then:
$$F=-[N / \beta] \log Z$$
We can also explore the standard method of calculating state functions from the free energy: ${ }^{9}$
\begin{aligned} &P=-(\partial F / \partial A){T} \ &S=-(\partial F / \partial T){A} \end{aligned}
or, using:
$$S=-k \beta^{2}(\partial F / \partial \beta)_{A}$$