ENTROPY IS THE COUNT OF WHAT YOU DO NOT KNOW
1. WHAT ENTROPY IS
Entropy is a number. This number counts the possible states of a system.
A system has two levels. The large level gives the quantities that you can measure: temperature, pressure, volume. The small level gives the position and the speed of each particle.
Many different small states give the same large state.
Entropy counts these small states.
The formula is S = k log W, where W is the count.
2. ENTROPY AND INFORMATION
Entropy also measures missing information.
You know the large state. You do not know the small state.
Entropy gives the quantity of data that you must get to find the small state. The unit is the bit.
Information is physical. When a machine erases one bit, the machine must release a minimum quantity of heat. This is the Landauer limit. Thus thermodynamic entropy and information entropy are the same quantity, in different units.
3. ENTROPY AND TIME
Entropy gives a direction to time. The entropy of a closed system does not decrease. This is the second law. The past is the direction of low entropy. The future is the direction of high entropy.
The universe started in a state of very low entropy. The gravitational field was smooth at the start. A smooth gravitational field has low entropy. This is the Weyl curvature hypothesis of Penrose. Because the start was low, entropy can increase. This increase is the arrow of time.
4. ENTROPY AND SPACE
A black hole has entropy. The entropy is equal to the area of the horizon, divided by 4, in Planck units. The area sets the entropy, not the volume. This is a special result. For usual matter, entropy increases with the volume.
This result gives the holographic principle. The surface area of a region sets the maximum information in that region.
Spacetime itself is possibly made from information. In 1995, Jacobson applied the laws of heat to small horizons. The result was the Einstein equations. In this view, spacetime is not fundamental. Spacetime is the large-scale shape of connected quantum information. The connection is entanglement. More entanglement between two regions makes the distance between them smaller.
5. SUMMARY
Entropy counts the states that you do not know. Information is physical, and area sets its limit in space. Time gets its direction from the increase of entropy, from a smooth start. Spacetime is possibly the geometry of quantum information. One quantity connects heat, time, space, and knowledge: entropy.
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