Order & Disorder: The Story of Information - Maxwells' Demon

https://www.youtube.com/watch?v=11QkX4u6RJg

Two advances would be crucial to solving Maxwell’s demon.

The first was by the American mathematician Claude Shannon, regarded as the founder of information theory. In 1948, Shannon showed that the information content of a message could be quantified with what he called the information entropy. “In the 19th century, no one knew about information,” said Takahiro Sagawa, a physicist at the University of Tokyo. “The modern understanding of Maxwell’s demon was established by Shannon’s work.”

In 1982, the American physicist Charles Bennett put the pieces of the puzzle together. He realized that Maxwell’s demon was at core an information-processing machine: It needed to record and store information about individual particles in order to decide when to open and close the door. Periodically it would need to erase this information. According to Landauer’s erasure principle, the rise in entropy from the erasure would more than compensate for the decrease in entropy caused by the sorting of the particles. “You need to pay,” said Gonzalo Manzano, a physicist at the Institute for Quantum Optics and Quantum Information in Vienna. The demon’s need to make room for more information inexorably led to a net increase in disorder.

Then in the 21st century, with the thought experiment solved, the real experiments began. “The most important development is we can now realize Maxwell’s demon in laboratories,” said Sagawa.

https://www.quantamagazine.org/how-maxwells-demon-continues-to-startle-scientists-20210422/

The Anti-Demonic Rebuttal:
Those that would argue against the existence of Maxwell's Demon will be quick to point out that examples cited while appearing to break the second law locally, do indeed obey the law globally. They will say yes, entropy is lowered locally by the photovoltaic cell, but this is more than offset by the increase of entropy in the Sun, created by nuclear fusion.

In other words, they would redraw the boundary of the experiment to include the Sun, thereby avoiding any paradox between their view point and the experimental result.

In the case of the laser, a similar redrawing of the boundary to include the power plant supplying the laser alleviates the paradox.

All fine and good, however… Why not adopt the same redefinition of boundary with respect to Maxwell's original demon hypothesis? Where did the kinetic (thermal) energy of the air molecules residing in the container come from in the first place, if not from the Sun?

Class dismissed

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Last week, we covered the scientific and societal controversy surrounding Maxwell's demon, and briefly touched on two devices that exhibit anti-entropic behavior.

In part 2 we shall take an in-depth look at the mechanism responsible for anti-entropic behavior in the laser. This mechanism is called a population inversion, and until the mid twentieth century, was considered little more than a quaint mathematical curiosity, with no basis in physical reality.

In order to lay a proper foundation for discussion of inverted populations, a short review of classical thermodynamics is included.

https://t.me/AzazelNews/243711

Degrees of Freedom:

At it's most basic level, the science of thermodynamics is a statistical study of vibration and movement in populations of atoms or molecules. In order to numerically quantify a population, it is necessary to understand how many different ways or modes of vibration and/or movement are available to the population. For instance, a population of molecules that have magnetic properties, when under the influence of a magnetic field, may be constrained to movement in a single direction, and yet in the absence of a magnetic field, this same population may have nearly unlimited directions of movement. In thermodynamics we use the term "degrees of freedom" to describe the number of available modes of vibration and/or directions of movement. Alternately, degrees of freedom can be viewed as the set of locations available to a population of atoms or molecules when it moves or vibrates. This set of available locations is called the population's "phase space".

Temperature, energy, and entropy:

The concept of temperature arose from a need to quantify the human perception of hot and cold. However at the beginning of the 19th century, no scientist was sure what temperature was really measuring. It was well known that expending energy (doing work), such as drilling a hole, or bending sheet metal caused the material to get hot, so heat was related to energy in some manner. But what was heat? Some believed it was a discrete substance, separate from matter (they called it caloric), while others believed it was an inherent property of matter, like mass or volume.



Boltzmann and the Bridge Between Two Worlds
https://www.youtube.com/watch?v=uJGyTcehW18

Boltzmann's controversial equation, Eq. 1 relates the entropy of an atomic or molecular population to it's phase space (disorder). As a side note, this equation was engraved on his tombstone...

The utility of Eq. 1 is that it defines entropy (disorder) as the ratio of energy to temperature.

Looked at another way, given a population with a constant energy content, as entropy declines, temperature rises.

In other words, as the degrees of freedom available to a population diminish, those few degrees of freedom still available, MUST contain the entire energy content of the population, and this causes the temperature of the population to rise.

Eq. 2a

Eq. 2b

Where:
S = Entropy (disorder).
Q = Energy in Joules.
T = Absolute temperature in degrees Kelvin

Eq. 2b implies that for any population with a non-zero energy content, as entropy approaches zero (one degree of freedom), the temperature of the population approaches infinity.

Maximum disorder:
Is there an upper limit to disorder? The answer to this question depends on the degrees of freedom available to a population of atoms or molecules. For most populations, and in the most general sense of the question, the answer is "no". This answer has a practical consequence. It implies there is no upper limit to temperature, since there will always be another dimension in phase space (degree of freedom), to which we can add another increment of energy, and thereby raise the temperature of the population.

However, if a population has a limited set of dimensions in phase space (limited degrees of freedom), there is a definite upper limit to disorder.

Surprisingly, the limit will be reached when exactly half of the dimensions in phase space are occupied by the population. It is obvious that when a population of atoms or molecules contain no energy, they do not vibrate or move, and therefore all members of the population occupy a single point in phase space (no disorder).

Figure 1 - Temperature versus phase space occupancy

Now consider the opposite condition.

I.E. Every member of the population is vibrating or moving in every available degree of freedom (all dimensions of phase space are occupied). Again, all members of the population are exactly alike, and again there is no disorder.

Therefore maximum disorder (and entropy) is achieved when exactly 50% of the dimensions in phase space are occupied.

As a consequence of this remarkable situation Eq. 2b implies the temperature of a population at maximum disorder is infinite, and the temperature of any population where more than 50% of phase space is occupied is negative AND hotter than infinity. Figure 1 shows the relationship between temperature and phase space occupancy for a population with limited degrees of freedom.

Quantum populations:
While classical physics allows populations of atoms or molecules to have nearly unlimited modes of vibration and movement (with corresponding degrees of freedom), quantum electrodynamics is a different story altogether.

Consider the electron orbits of a Hydrogen atom. The allowed orbital values are discreet, and defined by Eq. 3. Therefore electron orbits represent a population with very limited degrees of freedom (dimensions in phase space).