When we also define a characteristic temperature, $\Theta_{\mathrm{E}}$, the Einstein-temperature, via $$ \Theta_{\mathrm{E}} \equiv \frac{h v_{\mathrm{E}}}{k_{\mathrm{B}}} $$ We get the following expressions for the molar energy of the crystal and it’s specific heat. and $$ \begin{gathered} \widetilde{E}^{\mathrm{E}}=\frac{3}{2} R \Theta_{\mathrm{E}}+\frac{3 R \Theta_{\mathrm{E}}}{\exp \left(\frac{\Theta_{\mathrm{E}}}{T}\right)-1} \ \widetilde{C}{V}^{\mathrm{E}}=\left(\frac{\partial \widetilde{E}^{\mathrm{E}}}{\partial T}\right){V}=3 R \frac{\left(\frac{\Theta_{\mathrm{E}}}{T}\right)^{2} \exp \left(\frac{\Theta_{\mathrm{E}}}{T}\right)}{\left[\exp \left(\frac{\Theta_{\mathrm{E}}}{T}\right)-1\right]^{2}} \end{gathered} $$
The density of frequencies, $N(v) \mathrm{d} v$, can be deduced from the density of states $$ N(k) \mathrm{d} k=\frac{k^{2}}{2 \pi^{2}} V \mathrm{~d} k \quad \text { with } k=\frac{2 \pi}{\lambda} \quad \text { and } v \lambda=c $$ Hence, with $k=\frac{2 \pi}{c} v$ and $\mathrm{d} k=\frac{2 \pi}{c} \mathrm{~d} v$ $$ N(v) \mathrm{d} v=\frac{4 \pi^{2} v^{2}}{2 \pi^{2} c^{2}} V \frac{2 \pi}{c} \mathrm{~d} v=\frac{4 \pi V v^{2}}{c^{3}} \mathrm{~d} v $$
Modern engineering systems are increasingly complex and include many interdependent and dynamic parts. This can allow us to develop a new framework to handle large amounts of data and provide real-time control actions to maximize the benefits involved. However, as we move to increasingly complex systems, we need to develop new decentralized control methods to optimize the impact of the interactions between their entities on the behavior of the system.
Centralized stochastic control has been the prevalent method for controlling complex systems with stochastic uncertainty. Centralized stochastic control is the multi-stage optimization problem of a system with external disturbances and noisy observations by a single decision maker. A key assumption in deriving solutions to centralized stochastic control problems is that the decision maker has complete memory of all past control actions and observations.
A complex system refers to a system in which there is no precise relationship between the results of that system and the original causes of those results. The main characteristic of a complex system is its unpredictable and non-linear dynamics. This system complexity is due to the intricate and heterogeneous coupling between the components of the system, which makes it impossible to analyse the components individually and isolate them from the rest of the system.5 The close coupling and interactions between the units of a complex system lead to collective behaviour that is identifiable on a larger scale.
When there are some neighbors in zone of repulsion $\left(n_{p} \neq 0\right)$, the individual $i$ only reacts with respect to them. As a result, the desired direction $w_{i}(t+T)=w_{r}(t+T)$ can be quantified from equation $(1)$ and equation $(2)$. If there is no individual in the zone of repulsion, then the desired direction will be defined based on neighbors in zone of orientation and attraction $\left(w_{i}(t+\tau)=\frac{1}{2} \times\left(w_{o}(t+\tau)+w_{a}(t+\tau)\right)\right)$. $w_{o}(t+\tau)$ and $w_{a}(t+\tau)$ can be quantified from equation (3) and equation (4). $$ \begin{aligned} &w_{a}(t+\tau)=\sum_{j=1}^{n_{a}} \frac{d_{j}(t)}{\left|d_{j}(t)\right|} \ &w_{a}(t+\tau)=\sum_{j=i}^{n_{a}} \frac{r_{i j}(t)}{\left|r_{i j}(t)\right|} \end{aligned} $$ Considering the desired direction vector at each time step, if $w_{i}(t+\tau)$ is less than maximum turning rate $\theta$, then $d_{i}(t+\tau)=w_{i}(t+\tau)$. On the other hand, if desired direction vector exceeds the maximum rate, then the individual rotates by angle of $\tau \times \theta$ towards the desired direction.
Free energy landscape. Our framework generalizes the method presented by Akinori Baba and coworkers $^{13,47}$ and constructed the strategy to estimate the free energy landscape for a group of $N$ agents moving in a three-dimensional space. In the following, we provide a brief overview of the procedure we used to identify and extract the states from time series of agents in the group. First, we divide the time series containing the location of all the agents denoted by $r(t)$ to sub-intervals centered at time $t_{c}$ with time window $\left[t_{c}-\Delta / 2, t_{c}+\Delta / 2\right]$, where $\Delta$ is the preferential time scale (Fig. 1a). In the next step, we construct the probability density function of the location of all the agents in the group corresponding to each sub-interval (i.e. $p_{i}$ ) and based on that we find cumulative distribution function (CDF) of the agents’ location in the space. We also estimate the CDF corresponding to the position for the entire group through the whole time in the same way. Based on Kantrovitch distance $d_{K}$ we compare the CDF of sub-intervals with whole time series CDF and cluster the sub-intervals based on the similarities (equation (5) $)^{58}$. $$ d_{K}\left(p_{i} | p_{j}\right)=\int_{-\infty}^{\infty}\left|\int_{-\infty}^{r}\left(p_{i}\left(r^{\prime}\right)-p_{j}\left(r^{\prime}\right)\right) d r^{\prime}\right| d r $$ We consider each of the clusters as a spatio-temporal state for the group dynamics (Fig. 1b). We calculate the escape time of each state, meaning the time between when the system enters and leaves each cluster.
We calculate the residential probability $P_{i}$ of the ith state and transition probabilities $P_{i j}$ from the $i$ th state to the jth state (Fig. 1c). Based on these probabilities, we estimate the free energy landscape by quantifying the energy level in each state $\left(F_{i}\right)$ from equation (6) and energy barrier for the group while evolving from state $i$ to state $j\left(F_{i j}\right)$ from equation $(7)^{47}$. $$ \begin{gathered} F_{i}=-k_{B} T \ln \left(P_{i}\right) \ F_{i j}=-k_{B} T \ln \left(\frac{h}{k_{B} T} P_{i j}\right) \end{gathered} $$
