By Jordan G. Brankov
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Extra info for Introduction to the Finite-Size Scaling
Example text
Bifurcation diagram of the rotating nonlinear pendulum oscillator, where ~ =n 2L/g So we see that beyond a certain critical value of the angular velocity n of the frame, the traditional stable steady state 8 =0 becomes unstable, and the new (symmetrical) stable steady-state solutions are given by 8 = arc cos~-l. Of course, we may as well reduce the preceding analysis to the motion of a heavy particle in a potential well. What is the potential function now? The external force acting on the system is F =w6 sin (~COS8 -1).
This is not always the case, however. In order to pass judgement we have to exami ne carefully the "cartography" of the state space around each such new state. , more accessible) candidates than others for receiving the system which, after being expelled from the former state, requires a "landing". In short, we have to examine the "stream1ines" around the available steady states in state space, in the hope of predicting the 1i ke 1i hood of the next state. 13 -perpendicular to the paper, and then looking "down" to appreciate the road system of the landscape.
E(I, ... , Il), ~i=l Ni =N. The number of microscopic arrangements or compZexions responsible for the same macrostate of the system is, of course, W= N! 15) i' If all complexions are a priori equiprobable, then the probability of having one of them responsible for the given macrostate is P =W- 1. Question: What is the uncertainty-and the information to be deduced there from-about which complexion is responsible for the observed macros tate? Think of the system under consideration as divided into two parts 1 and 2.