Question

If you roll 1 million dice, what will be the average of all the dice? If there is a room with 1 million dice and they all have a 1 on the top face, and there is an earthquake strong enough to roll dice around, what is the likelihood that after the earthquake all the top faces will sum to 1 million? To 6 million? How does this thought experiment illustrate the second law of thermodynamics?

Short answer

The average roll of 1 million dice is 3,500,000. The likelihood of them all showing '1' or '6' after an earthquake is \(\left(\frac{1}{6}\right)^{1,000,000}\) respectively, illustrating that a highly ordered state is extremely improbable, analogous to the second law of thermodynamics which predicts systems tend to increase in entropy.

Step-by-step solution

Understanding Dice Average

The average roll of a single die, with faces numbered 1 to 6, is computed by taking the sum of the possible outcomes and dividing by the number of possible outcomes. This is given by the formula for the average (mean) value of a uniform distribution: Average = (1 + 2 + 3 + 4 + 5 + 6) / 6.

Calculating Single Dice Average

Calculate the average of a single dice roll: \[ \text{Average} = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{21}{6} = 3.5 \]

Applying Average to 1 Million Dice

Since each die is independent, the average for 1 million dice is the same as the average for a single die. Therefore, if you roll 1 million dice, the expected sum is 1 million times the average of a single die: \[ 1,000,000 \times 3.5 = 3,500,000 \]

Probability of All Dice Showing '1'

The probability of a single die landing on '1' is 1 out of 6. Since the dice are independent, the probability for 1 million dice all showing '1' is: \[ \left(\frac{1}{6}\right)^{1,000,000} \]

Probability of All Dice Showing '6'

Similarly, the probability of a single die landing on '6' is also 1 out of 6. The probability for 1 million dice all showing '6' is the same as the probability of them all showing '1': \[ \left(\frac{1}{6}\right)^{1,000,000} \]

Connecting to Second Law of Thermodynamics

The thought experiment illustrates the second law of thermodynamics, which states that systems tend to move towards disorder or entropy. It's extremely unlikely (though not impossible) for all the dice to show '1' or '6' indicating a highly ordered state. After an earthquake, the natural tendency is for the dice to land in a variety of positions, not all in the one specific ordered state, resembling the increase in entropy. This increase in entropy is analogous to the natural degradation of energy in a thermodynamic system.

Key concepts

Entropy

Imagine tossing a coin. You have an equal chance of it landing on heads or tails. Now picture a million coins flipped into the air; some might guess that they should have about half a million heads and half a million tails. But life, like thermodynamics, often defies such exact expectations due to randomness. Entropy is a measure of this randomness or disorder within a system.

In our dice example, entropy represents the number of different ways the dice can land. Having all dice land on the same number, say all ones or all sixes, reflects a state of low entropy, an ordered state. The chances of this happening are astronomically low. In contrast, the high entropy state is where the dice land in a mix of numbers from one to six, which is not only more likely but also more disordered.

The second law of thermodynamics tells us that the total entropy of an isolated system can never decrease over time. In other words, the natural progression of any isolated system is to move towards a state of maximum disorder. This randomness or statistical spread is what makes the outcome of every dice roll (and other complex systems) unpredictable over time.

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random phenomena and predicting the likelihood of various outcomes. It provides the backbone for understanding the dice problem. The probability of rolling a one on a single die is one out of six. But what happens when we roll a million dice?

Here's the crux: the probability of all million dice showing a '1' can be found by multiplying the chance of a single die landing on '1', raised to the power of one million, as shown by the remarkably small number \( (1/6)^{1,000,000} \). It's akin to the likelihood of flipping a coin and getting heads a million times consecutively – nearly impossible.

Probability theory isn’t just about one-time events; it also takes into account the average outcomes over many trials. The expected average of a single die is 3.5, a predictable outcome in the long run across many rolls, due to the law of large numbers which states that as more trials are performed, the experimental probability of an event tends to get closer to the theoretical or expected probability.

Statistical Mechanics

Statistical mechanics is a framework that links the macroscopic properties of materials to the microscopic behaviors of atoms and molecules. It’s the bridge between the laws of thermodynamics and the erratic world of individual particles.

Our dice, like molecules in a gas, exhibit random behavior. But statistical mechanics allows us to calculate macroscopic properties - like pressure or temperature for a gas - from the statistical distribution of microscopic states. In dealing with a million dice, statistical mechanics helps us predict the overall sum or distribution from the individual probabilities of each die. This is analogous to how thermodynamic properties emerge from the movements and interactions of particles.

In essence, statistical mechanics tells us that while we can't predict the exact landing of each die, we can accurately predict the average behavior of the dice ensemble. This average, resulting from countless microscopic events, brings a form of predictability to systems that seem fundamentally unpredictable - a core aspect of why we observe stable laws of physics in our macroscopic world.