Question

Probability and Thermodynamics: Analysing feasibility of 50 benches problem.

I have a thought experiment that involves dropping 50 benches from a height through an open roof and examining the probability of them spontaneously arranging themselves into neat rows and columns. This scenario raises questions at the intersection of mathematics and physics. Here's the setup:

Imagine an idealized scenario where there are:

50 identical benches.
An isolated system with no dissipative forces.
A closed and perfectly athermic enclosure.
A defined temperature for the system.
Given these conditions, the probability of the benches forming ordered rows and columns seems to defy our intuition and the second law of thermodynamics.

Here are my questions:

Mathematical Feasibility: Can we mathematically calculate the probability of the benches arranging themselves into ordered rows and columns under these idealized conditions, and if so, what mathematical tools or concepts should be employed?

Physical Realism: From a physics perspective, is this scenario truly feasible, or does it run counter to established principles like the second law of thermodynamics? How do we reconcile the apparent violation of this law in this thought experiment?

Limits of Idealization: To what extent do the idealized conditions imposed on this scenario (perfect isolation, closed athermic walls, etc.) affect the theoretical feasibility, and how does this relate to the real-world limitations and observations?

Answer 1

Mathematically, the probability of 50 benches spontaneously arranging themselves is virtually zero, and physically, the second law of thermodynamics makes the scenario extremely unlikely due to the natural increase in entropy of isolated systems.

Step-by-step explanation:

The question of arranging 50 benches into neat rows and columns through a random process involves both the mathematical concept of probability and the physical principle of thermodynamics.

Mathematically, one could calculate the probability using combinatorics to determine the number of arrangements (microstates) that result in the organized pattern versus all possible arrangements. However, this number would be so astronomically small that for all practical purposes it is zero.

From a physics standpoint, the second law of thermodynamics indicates that in an isolated system, such as the one described, entropy, or disorder, will tend to increase, not decrease.

The scenario described essentially requires a decrease in entropy, which, while not outright impossible, is highly improbable - akin to witnessing a significant spontaneous decrease in entropy like melted ice spontaneously refreezing.

The idealized conditions such as perfect isolation do not exist in reality and even hypothetical perfectly isolated systems follow statistical mechanics where spontaneous decreases in entropy are exceedingly unlikely. This aligns with real-world observations where entropy typically increases over time.