Trajectories symmetry breaking

Figure 3 depicts the spectmm of Lyapunov exponents in a hard-sphere system. The area below the positive Lyapunov exponent gives the value of the Kolmogorov-Sinai entropy per unit time. The positive Lyapunov exponents show that the typical trajectories are dynamically unstable. There are as many phase-space directions in which a perturbation can amplify as there are positive Lyapunov exponents. All these unstable directions are mapped onto corresponding stable directions by the time-reversal symmetry. However, the unstable phase-space directions are physically distinct from the stable ones. Therefore, systems with positive Lyapunov exponents are especially propitious for the spontaneous breaking of the time-reversal symmetry, as shown below. 

The time evolution of the probability density is induced by Hamiltonian dynamics so that it has its properties—in particular, the time-reversal symmetry. However, the solutions of Liouville s equation can also break this symmetry as it is the case for Newton s equations. This is the case if each trajectory (43) has a different probability weight than its time reversal (44) and that both are physically distinct (45). 

On the other hand, the nonequilibrium steady states are constructed by weighting each phase-space trajectory with a probability which is different for their time reversals. As a consequence, the invariant probability distribution describing the nonequilibrium steady state at the microscopic phase-space level explicitly breaks the time-reversal symmetry. 

The photodissociation of symmetric triatomic molecules of the type ABA is particularly interesting because they can break apart into two identical ways ABA — AB + A and ABA — A + BA. Figure 7.18(a) shows a typical PES as a function of the two equivalent bond distances. It represents qualitatively the system IHI which we will discuss in some detail below. We consider only the case of a collinear molecule as illustrated in Figure 2.1. The potential is symmetric with respect to the C -symmetry line 7 IH = i HI and has a comparatively low barrier at short distances. The minimum energy path smoothly connects the two product channels via the saddle point. A trajectory that starts somewhere in the inner region can exit in either of the two product channels. However, the branching ratio ctih+i/cti+hi obtained by averaging over many trajectories or from the quantum mechanical wavepacket must be exactly unity. 

If the quantum corrections to conductivity are actual the magnetoresistance related to the influence of the magnetic field on these corrections takes place [57-59]. The interference of electrons passing the closed part of trajectory in clockwise and counter-clockwise directions causes the so-called corrections to the conductivity. The phases of the electron wave functions in this case are equal and so this interference is constructive. Therefore, the probability for electrons to come back to the initial point doubles. This leads to the interference corrections which increase the classical resistance. The external magnetic field breaks the left-right symmetry, and the phases collected by the electron wave function while it passes trajectory in clockwise and... 

Amazingly, most trajectories recross the TS, nsually by reaching into the region near the second TS. However, the recrossing decreases with increasing isotopic mass. Vibrational mode 3 breaks the symmetry movement in one direction along... 

---