Entropy

Entropy is the tendency of everything to trend toward greater disorder, in the absence of intelligent intervention.

The second law of thermodynamics states that entropy will never decrease over time within a closed system, defining a closed system as one in which neither matter nor energy may enter or leave.[1]

Entropy is undeniable and yet creates perhaps insurmountable difficulties for many modern theories of physics. For example, it renders time asymmetric, resulting in an arrow of time that is impossible to reconcile with the theory of relativity. Entropy casts doubt on whether physical laws or the speed of light are invariant and perpetual. Increasing entropy renders the theory of evolution implausible, because that theory claims that order is increasing. Liberal denial is thus common in ignoring the significance of the increase in disorder.

Definitions

Thermodynamic definition

In classical thermodynamics, if a small amount of energy dQ is supplied to a system from a reservoir held at temperature T, the change in entropy is given by

$dS=\frac{dQ}{T}$

For a measurable change between two states i and f this expression integrates to

$\Delta S=\int_{i}^{f}\frac{dQ}{T}$

Statistical mechanics definition 1 (Boltzmann Entropy)

If a system can be arranged in W different ways, the entropy is

S = kBlogW

where kB is Boltzmann's constant.

Statistical mechanics definition 2 (Gibbs Entropy)

Label the different states a thermodynamic system can be in by $i=1,2,3\ldots N$. If the probability of finding the system in state i is pi, then the entropy is

$S=-k_B \sum_{i=1}^N p_i \log p_i$

where kB is the Bolzmann constant. This definition is closely related to ideas in information theory, where the definition of information content is very similar to the definition of entropy.

Entropy in information Theory (Shannon Entropy)

For a discrete random variable, entropy is defined as

$H(X)=-\sum_{x\in \mathcal{X}} P(X=x)\log_2 \left(P(X=x)\right)$

For a continuous random variable, the analogous description for entropy, which in this case represents the number of bits necessary to quantize a signal to a desired accuracy, is given by

$h(X)=-\int_{x\in \mathcal{X}} f_X (x) \log_2 \left(f_X (x)\right)$

Entropy in Quantum Information Theory (Von Neumann Entropy)

In entangled systems, a useful quantity is the Von Neumann Entropy, defined (for a system with density matrix ρ by

$H(\rho) = -\bold{Tr} \left( \rho \ln \rho \right)$

where Tr() indicates taking the Trace of a matrix (the sum of the diagonal elements). This is a useful measure of entanglement, which is zero for a pure state, and maximal for a fully mixed state.