Maxwell-Boltzmann Statistics

Maxwell Boltzmann statistics is applicable to identical, distinguishable particles of any type of spin. The molecules of gas are particles of this type.

Particles regulated by Maxwell-Boltzmann Statistics have to be distinguishable, and one energy state can be occupied by two or more particles. The probability distribution is given by the formula:

The Boltzmann distribution is a probability distribution that gives the probability of a certain state as a function of that state's energy and temperature of the system.

Bose Einstein statistics  is applicable to identical, indistinguishable particles of zero or integral spin (Bosons, e.g., photon, He atom).

Particles regulated by Bose-Einstein Statistics have to be indistinguishable, and one energy state can be occupied by two or more particles. The probability distribution is given by the formula:

Fermi Dirac statistics  is applicable to identical, indistinguishable particles of half-integral spin (e.g., Electron, proton) obeying Pauli Exclusion Principle.

Particles regulated by Fermi-Dirac Statistics have to be indistinguishable, and one energy state can be occupied by only one particle.

Relationship between sums and  integrals

what are those conditions on "f"?

Boltzmann Machine as a Pontryagin Observer in Sensor Network

Einstein-Podolsky-Rosen paradox has co-founded physicists trying to understand how to interpret quantum measurements. "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?"

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Boltzmann Machine  is a generative unsupervised model. It involves learning a probability distribution from an original dataset and using it to make inferences about never-before-seen data.

Reconstruction is different from regression or classification as it estimates the probability distribution of the original input. The joint distribution is known in Physics as the Boltzmann Distribution. It gives the probability that a particle can be observed in the state with energy E.

This is where Restricted Boltzmann Machines (RBM) meet Physics. RBMs are probabilistic, assigning probabilities instead of discrete values. They estimate the probability distribution of the original input.

In the following example, there are 3 hidden units and 4 visible units. This is NOT a restricted Boltzmann machine. Each undirected edge represents a dependency.

In the above example, there are 3 hidden units and 4 visible units. This is a restricted Boltzmann machine.

Let E be Global Energy