section_path: Applications/Entropy and chaos
section_name: Entropy and chaos
paragraph_text: Entropy and chaos
  
  Entropy is broadly a measure of the disorder of some system. In statistical thermodynamics, the entropy S of some physical system is defined as S = − k ∑ i p i ln (p i). {\displaystyle S=-k\sum _{i}p_{i}\ln(p_{i}).\,} The sum is over all possible states i of the system in question, such as the positions of gas particles in a container. Moreover, p is the probability that the state i is attained and k is the Boltzmann constant. Similarly, entropy in information theory measures the quantity of information. If a message recipient may expect any one of N possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as log N bits.
  
  Lyapunov exponents use logarithms to gauge the degree of chaoticity of a dynamical system. For example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle. Such systems are chaotic in a deterministic way, because small measurement errors of the initial state predictably lead to largely different final states. At least one Lyapunov exponent of a deterministically chaotic system is positive.