# Uncertainty and Thermodynamics

## Dilemma

The uncertainty principle of energy and the 2nd law of thermodynamics don't add up : the uncertainty principle of energy says that

$\Delta \tau \cdot \Delta E \ge \frac{h}{4\pi} = \frac{\hbar}{2}$

where $\Delta$ is the uncertainty in measurement.

Now lets consider a situation: lets say that an isolated system $A$ is in thermodynamic equilibrium. It has two particles $b$ and $c$ so that it is in the state of maximum possible entropy. To preserve the uncertainty principle, some net energy must flow from $b$ to $c$ or from $c$ to $b$, and that results in a non-equilibrium state, may be for a fraction of a fraction of a second. But the system must move from the state of maximum entropy (equilibrium) to the state of lesser entropy, which is a sure violation of the second law of thermodynamics. Does anyone have an explanation?

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your question is really hard to read and quite hard to understand. why do you claim that to preserve the uncertainty principle, some net energy must flow from b to c or from c to b [... ]? could you elaborate your reasoning further? –  sebastian Apr 12 at 16:41
Decreasing entropy for a short period does not violate the second law of thermodynamics, since the law only governs the average entropy change. Moreover with such a small number of particles, the entropy isn't really definable in a useful way - what are the macrostates for example? –  twistor59 Apr 12 at 16:51
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## 1 Answer

@twistor59 has it basically right. Classical thermodynamics is a macroscopic subject and while some of its variables such as internal energy and volume can be defined on a microscopic level, many, such as temperature and pressure, really are collective effects of a large number of particles.

Entropy falls into this second class. Entropy is really defined as a macroscopic quantity and it makes little sense to speak of the entropy of a very few particles.

That said, statistical thermodynamics is a link between microscopic ideas and macroscopic ones. Using it one can sometimes define thermodynamic properties such as entropy for very small systems. However, when one mixes microscopic and macroscopic ideas strange things can result. In your situation how would you envision the uncertainty principle working experimentally?

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