Beyond Chaos
Entropy is a profound concept. The second law of thermodynamics may have given entropy the job of being the arrow of time, but it goes way further. Entropy is an important concept in information theory, genetics, economics, and more.
For example, in genetic and DNA sequencing, Shannon entropy became a standard measure. In economics, entropy enters again as many “thermo-economists” have combined these fields, including the concept of information theory and statistical physics. Chemical processes, either reversible or irreversible, are based the entropy of the macrostates into account. Similarly, these economists conjecture that the direction of economic change may involve the entropy of neighboring microstates.
Claude E. Shannon, the American Mathematician, was one of the first to take entropy out of its limited domain and into a very new field — informatics.
Forces and Energy
Most of the forces in our universe are connected, in some way, with energy.
If we drop a ball from a certain height, its gravitational potential energy causes it to fall down. If we hold opposite ends of magnets together, the poles are attracted due to the energy of the magnetic field. The poles were presented with a state having lower energy than the current state they existed in, so they moved in that way, exerting a force.
As a general rule in physics, objects tend to move from a state of higher energy to a state of a lower energy.
To go from the higher state to the lower one, they will exert a force. Thus, conservative forces are given by the change in potential as:
$$F = -\frac{dU}{dx}$$
But is it possible to have a force without an energy change?
The answer is yes.
Entropy and Energy
Let’s return entropy back to where it all started. The arrow of time. A property so close and so similar to energy, it might even substitute energy. Consider the case of the gas in the piston as shown below.
The gas within the piston exerts a force on the walls of the container. As the walls of the container are rigid, they don’t move. The piston however is pushed outwards by the gas. The internal energy does not depend on the volume, and the temperature of the gas has not changed. So where did the force needed to create this expansion come from? The answer is not energy, and is entropy.
Thermal fluctuations in a thermodynamic system bring the system towards the macroscopic state with the maximum possible entropy. Therefore, the entropic force is said to be a force that emerges as a result of a system’s statistical tendency to increase the entropy, rather than any forces at the atomic-scale.
But what exactly is an emergent phenomenon?
Consider a container with air inside it. The mass of the system can be measured as a whole. Yet the mass comes from the individual mass of the particles of the various gases which compose the air.
But now consider, for example, me writing this article. I need to think of the words and type them, while I also need a medium in which to put them. The process of me writing, which involves both the computer and me, is an emergent phenomenon, some property or process which can be done only by the system as a whole.
Another example is the saltiness of salt. The salty taste doesn’t come from the Na+ ions or the Cl- ions, it comes from them both, it is an emergent property of the system.
The elastic force of polymers also has entropic origin. When a polymer is stretched wide, the number of states the system can be in is reduced, since the string of molecules and bonds is pulled taut.
When the polymer is back in its normal state (relaxed and slack) it can be in many more states, which means higher entropy. And since the molecules in the polymer would prefer being in the higher entropy state, they naturally bounce back to their original shape.
But the strangest way an entropic force manifests itself is in the form of one of the fundamental forces of nature — gravity.
The gravitational force was described entropically in a model by Erik Verlinde in 2009. The thermodynamic description of gravity has a history that goes back all the way back to 1973 when Jacob Bekenstein and Stephen Hawking tried to explain the entropy needed by a black hole to comply with the laws of thermodynamics. This was taken further in 1995 when Theodore Jacobson showed that Einstein’s field equations describing relativistic gravitation can be combined with thermodynamics with the help of the equivalence principle - the equivalence of gravitational and inertial mass.
But what in the world is the difference between them in the first place for an equivalence principle to prove their equivalence? The inertial mass is very simple, it is the mass we’re so used to in most of our calculations - it is the measure of how fast an object accelerates given the same force. Gravitational mass, however, is a product of Newton’s Universal Law. It paved the way for the discovery of atoms in a way - that we’re all but stardust, made of the same atoms, the very same mass everywhere. The very same mass which causes a gravitational field. The equivalence principle states that they are equivalent (Euclid’s fourth postulate dabs left and right multiple times). Another example is as follows. You’ve all probably heard this analogy sixty nonillion times before, but anyway, consider you are in an elevator. It’s blocked from the outside world (of course) and it’s freely falling. It is impossible to tell whether you are just standing in the middle of the Boötes void (standing in a virtually gravitation-free space) or trying to make another happy landing on Coruscant (freely falling).
Back to the main point, black holes, a gravitational beast was being tamed by thermodynamic equations, entropically. This sparked the idea that gravity and entropy are related. While the equivalence principle was the tool to pick out the black hole entropy with the second law there, here the holographic principle is the tool to dissect gravity entropically.
Now, what is this new principle?
The holographic principle itself was inspired by black hole thermodynamics. It is a fundamental pillar of string theories and is a supposed property of quantum gravity. It was first proposed by Gerard ‘t Hooft. Leonard Susskind gave it a string theory interpretation by combining ‘t Hooft’s ideas with his own. It states that the description of a volume of space can be thought of as entropically encoded as information on a lower dimensional boundary to the region. Something like the 4D volume of a surface encoded on its 3D volume, loosely speaking, or encoding the 3D volume on its 2D boundary, its surface area.
Let’s go along a tangent. The equipartition theorem is a pretty important part of classical, statistical thermodynamics. Systems have multiple states they can be in and the set of all states constitutes its phase space. The degrees of freedom of the system are the dimensions of the phase space so generated. The equipartition theorem states that the energy of the system is equally divided between its degrees of freedom.
Let’s go along a normal. The Unruh effect is a weird prediction of the quantum field theory. From the point of an accelerating observer, empty space (vacuum) emits black body radiation at a temperature proportional to the acceleration. To put it very simply, if you measure the temperature in a proper vacuum while in an accelerating frame, the thermometer would show a non-zero reading. The Unruh temperature also has the same form as the Hawking temperature of a black hole. Hence, it is also called as the Hawking-Unruh temperature. This effect may seem rather counter-intuitive. But bear in mind that vacuum, in modern terms, is not defined as space devoid of matter, but rather as the lowest possible energy of the quantized fields that make up the universe.
Now let’s put it all together.
Let’s start from the part we got from the holographic principle. Let us encode $N$ bits of information into an area, $A$. The Planck Length is given by $l_P$
$$N = \frac{A}{{l_P}^2}$$
Also, from the equipartition theorem, we get,
$$E = \frac{1}{2}{k_B}NT$$
From the Unruh effect,
$$T = \frac{\hbar a}{2 \pi c k_B}$$
Now, as we are doing this derivation classically, we take Newton’s Second Law,
$$F = ma$$
Putting in a rearranged form of the Unruh equation,
$$F = m \times \frac{2 \pi c k_B T}{\hbar}$$
Substituting the equipartition theorem,
$$F = \frac{4 \pi c E}{N \hbar}$$
From the mass-energy equivalence principle,
$$F = \frac{4 \pi c^3 M}{\hbar N}$$
Putting in the encoded information and substituting the standard formula for Planck length,
$$
$$F = \frac{4 \pi GMm}{A}$$
If we take the holographic screen as a sphere of radius of $r$, and put it’s surface area,
$$F = \frac{GMm}{r^2}$$
Hey, now doesn't that look familiar!
But alas, for beauty can only take you so far. We must also test for accuracy as much as beauty. Though entropic gravity has been able to, in its current form, reproduce the Einstein field equations, it would have to be tested at the Lagrange points - the points where gravitational forces between two bodies are balanced. The theory has also been challenged on other formal grounds. Nevertheless, it still shows how physics can truly be beautiful.