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Indivisible particles: How does the quantum world solve the Gibbs paradox?
In the field of statistical mechanics, the Gibbs paradox has aroused in-depth thinking in the scientific community about the nature of entropy and its relationship with particle distinguishability since it was proposed. Between 1874 and 1875, the famous physicist Josiah Gibbs proposed a thought experiment related to the entropy of an ideal gas, demonstrating the paradoxical consequences that arise when particle indistinguishability is not taken into account: The entropy of the system may decrease, which violates the second law of thermodynamics.
"The core of this paradox is that if the entropy of an ideal gas is not an extended property, then the sum of the entropies of two containers of the same gas is no longer simply double."
Gibbs's way of thinking involves a simple assumption: two identical containers of ideal gases, when the partition between them is removed, the gases will mix naturally, but if the entropy expression is not extensive, then The entropy of the system after mixing will not be 2S, and may even exceed it. This is complicated by the distinguishability of the gas particles, and when the partition is reinserted, some properties of the system return to their previous state, showing a decrease in entropy, which is a clear violation of the principles of thermodynamics.
The reason for this paradox is closely related to the definition of particles. Gibbs's non-extensive entropy is not applicable to situations where the number of particles changes, without considering the distinctiveness of the particles. This paradox is resolved when it is assumed that the gas particles are effectively indistinguishable, leading to the Sackur–Tetrode equation for the extended properties.
Calculation of Entropy and Its Extension
The calculation of the entropy of an ideal gas involves the description of the particles in phase space. Assume an ideal gas containing N particles, with internal energy U and volume V. By describing the position vector and momentum vector of each particle, we can describe the state of the system. However, this process follows the assumptions of classical thermodynamics, which treats the states of particles as distinguishable.
"When the entropy of an ideal gas of N particles is calculated, the result of classical physics is infinite, while quantum mechanics provides a finite explanation."
In classical physics, the number of states is infinite, but the introduction of quantum mechanics allows this calculation to be revised in the semiclassical limit. According to Heisenberg's uncertainty principle, certain regions of the state space cannot be specified explicitly. This can cause some computational problems: if the energy specified is not accurate, this can cause the entropy calculation to diverge.
The hybrid paradoxThe paradox of mixing is closely related to the Gibbs paradox. When considering the mixing of two gases of different properties, the resulting entropy change does not depend solely on the ordering of their particles, but is based on the distinctiveness of the two gases themselves. This means that if the gases are mixed together and are identical, their entropy will not increase, and this phenomenon has led to intensive research on the definition of entropy.
"In theory, the classification of gases may be arbitrary, and the definition of entropy is to some extent a subjective judgment."
According to Edwin Thompson Jaynes, the definition of entropy is variable, meaning that more detailed measurements of the properties of a gas may change its definition. The importance of this in scientific research is that the increase or decrease in entropy clearly highlights the critical impact of indistinguishability in quantum mechanics on entropy calculations.
Strategies for resolving the Gibbs paradox
Finally, understanding the Gibbs paradox and its related concepts is crucial for furthering research in thermodynamics and quantum physics. By properly taking into account the indistinguishability of particles and using the Sackur–Tetrod equation, we are able to transform the calculation of entropy into a formula for extensive mass. This not only solves the Gibbs paradox, but also guides the direction of future thermodynamic research.
"In the quantum world, the indistinguishability of particles is not only a property, but also the key to understanding entropy and its transformations."
To this point, the study of Gibbs's paradox and its interaction with quantum theory has deepened our understanding of entropy, and all of this has raised an important question: How do we rethink entropy within the framework of quantum mechanics? How to define the nature and calculation of entropy?
