Tru Physics

Entropy

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Tru Physics
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Introduction

Entropy is a fundamental concept in thermodynamics, often interpreted as a measure of disorder in a system. It is a state function that describes the number of microscopic configurations (microstates) that a thermodynamic system can have when in a state of thermodynamic equilibrium.

Definition

Entropy (S) is defined for an isolated system in the thermodynamic limit, and its change (dS) is given by the reversible heat (\delta Q_{\text{rev}}) divided by the absolute temperature (T):

dS = \dfrac{\delta Q_{\text{rev}}}{T}

This equation connects the macroscopic thermodynamic description with the underlying microscopic description.

Boltzmann’s Entropy Formula

Boltzmann’s entropy formula relates the macroscopic state (characterized by macroscopic state variables such as temperature, volume, and number of particles) to the microscopic state (characterized by the position and momentum of every particle in the system). The formula is given by:

S = k_B \ln(\Omega)

where k_B is the Boltzmann constant, \ln is the natural logarithm, and \Omega is the number of microstates corresponding to the given macrostate.

Second Law of Thermodynamics

The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. It can increase, and will do so for any irreversible process.

Entropy in Information Theory

In information theory, entropy is used to quantify information, uncertainty, or surprise. The concept was introduced by Claude Shannon in his 1948 paper “A Mathematical Theory of Communication”. Shannon entropy is defined for a discrete set of probabilities p_i as:

H = - \sum_i p_i \ln(p_i)

This measure of entropy can be interpreted as the average amount of “surprise” or “information” per event, when the events are drawn from the given distribution.

Conclusion

Entropy is a central concept in both thermodynamics and statistical mechanics, with wide-ranging applications in physics, chemistry, and information theory. It encapsulates the fundamental nature of physical processes, from the cooling of hot objects to the operation of heat engines, and underpins the flow of time itself.

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