## 3 Phase Transition-Based Optimization Algorithm
### The Overall Design of the PTBO Algorithm
#### 3.3.2 Iterations of the Three Operators  
Now we simply give some certain implementation details about the three operators in the three different phases.  
**(1) Stochastic operator**  
Stochastic diffusion is a common operation in which elements randomly move and pass one another in an unstable phase. Although the movement of elements is chaotic, it actually obeys a certain law from a statistical point of view. We can use the mean free path [43], which is a distance between an element and two other elements in two successive collisions, to represent the stochastic motion characteristic of elements in an unstable phase. Figure 4 simply shows the process of the free walking path of elements.  
Figure 4: Two-dimensional example showing the process of the free walking path of elements.  
Figure 3: A phase transition from an unstable phase to a stable phase.  
The free walking path of elements is the distance traveled by an element and other individuals through two collisions. Therefore, the stochastic operator of elements may be implemented as follows:  
\[newX_{i}=X_{i}+rand_{1}*(X_{j}-X_{i})+rand_{2}*(X_{k}-X_{i}) \tag{2}\]  
where \(newX_{i}\) is the new position of \(X_{i}\) after the stochastic motion, \(rand_{1}\) and \(rand_{2}\) are two random vectors, where each element is a random number in the range (0, 1), and the indices \(j\) and \(k\) are mutually exclusive integers randomly chosen from the range between 1 and \(N\) that is also different from the indices \(i\).  
**(2) Shrinkage operator**  
In a meta-stable phase, an element will be inclined to move closer to the optimal one. From a statistical standpoint, the geometric center is a very important digital characteristic and represents the shrinkage trend of elements in a certain degree. Figure 5 briefly gives the shrinkage trend of elements towards the optimal point.  
Hence, the gradual shrinkage to the central position is the best motion to elements in a meta-stable phase. So, the shrinkage operator of elements may be implemented as follows:  
\[newX_{i}=newX_{i}+(X_{gb}-X_{i})*N(0,1) \tag{3}\]  
where \(newX_{i}\) is the new position of \(X_{i}\) after the shrinkage operation, \(X_{gb}\) is the best individual in the population, and \(N(0,1)\) is a normal random number with mean 0 and standard deviation 1. The Normal distribution is an important family of continuous probability distributions applied in many fields.  
**(3) Vibration operator**  
Elements in a stable phase will be apt to only vibrate about their equilibrium positions. Figure 6 briefly shows the vibration of elements.  
Hence, the vibration operator of elements may be implemented as follows:  
\[newX_{i}=newX_{i}+(2*rand-1)*stepSize \tag{4}\]  
where \(newX_{i}\) is the new position of \(X_{i}\) after the vibration operation, \(rand\) is a uniformly distributed random number in the range (0, 1), and \(stepSize\) is the control parameter which regulates the amplitude  
Figure 5: The shrinkage trend of elements towards the optimal point.  
Figure 6: The vibration of elements in an equilibrium position.  
of jitter with a process of evolutionary generation. With the evolution of the phase transition, the amplitude of vibration will gradually become smaller. \(stepSize\) is described as follows:  
\[stepSize=\exp{\left(1-G/(G-g+1)\right)}. \tag{5}\]  
where the \(G\) and \(g\) denote the maximum number of iterations and current number of iteration respectively, and \(\exp()\) stands for the exponential function.