It should be noted that the current ad hoc definition of the Gaussian distribution in quantum mechanics is one of the vaguest and most misleading in the history of useless R^4 mathematics.
"A discrete Gaussian distribution is a distribution on a fixed lattice where each lattice point is sampled with a probability proportional to its probabilistic mass, according to the standard Gaussian distribution (in n dimensions)".
It is clear that the above definition is vague and misleading and should be omitted and replaced with:
A discrete Gaussian distribution is defined as a stationary distribution of the time evolution of the interacting elements (binary interaction) of any closed system, regardless of its initial conditions.
The Gaussian distribution f(x) of the random variable x, parameterized by a mean (μ) and a variance (σ²).
The Gaussian distribution is not easy to derive mathematically in R^4 space, but it can be obtained simply in the 4-dimensional unitary x-t space using statistical techniques based on B matrices:
It should be noted that the transition matrix B here is defined by the author for the first time in 2020 and is completely different from the B matrix commonly used in other fields (which is obviously not a transition matrix).
f(x)dx = 1/ σ √2 π Exp -1/2( [x - μ ]/ σ)². dx. . . . According to the usual conventions.
We assume that it can be obtained simply numerically using modern statistical mechanics of B matrices, in the same way as statistical numerical integration.
The Gaussian distribution can be expressed as follows: f(x) = C1.Exp(-x^2/σ^2).
In fact, this is the Gaussian or normal distribution.
Furthermore, Boltzmann's famous law of entropy:
s=k log W
where k is Boltzmann's constant and s is the microscopic entropy, which coincides with the macroscopic entropy S of the second law of thermodynamics,
dS.GE.dQ /T.
It is clear that by proceeding simply, we can deduce Boltzmann's famous law of entropy:
s = k log W.
I cannot see anything like that
S=-kb ln (omega)
Omega is a volume in phase space at energy E
SIMPLE CHALLENGE
The simple challenge to our distinguished contributors and readers is the statistical transition matrix B for a closed control volume of surface A subject to Dirichlet boundary conditions,
U(x,y,z,t+dt)=B. U(x,y,z,t) . . . (1)
Equation 1 is not only universal, but it also describes and solves all time-dependent phenomena in the entire universe (partial differential equations of classical physics in its most general form, quantum physics, statistical distributions, integration and differentiation, etc.).
The challenge is as follows:
Name a single physical phenomenon from the four above that does not belong to Equation 1.
Thank you.
No. Boltzmann’s entropy formula is not a “subset” of a generalized Gaussian. The real link goes the opposite way: if you start from Boltzmann–Gibbs (or, in information-theory language, Shannon) entropy and maximize it subject to fixed mean and variance, the probability density that pops out is a Gaussian. Boltzmann’s law is therefore the generating principle; the Gaussian is one particular solution that obeys it under those constraints.
1 What Boltzmann actually wrote
Boltzmann’s microscopic entropy is
S=kB lnW,S
where W is the number of micro-states compatible with a macro-state. In the continuous limit this becomes the functional
S[f]=−kB∫f(x) lnf(x) dx,
subject to the usual normalisation and energy (or moment) constraints.
@This is mathematically identical to the Shannon (differential) entropy used in statistics and information theory.
2 Maximum-entropy ⇨ Gaussian
If you maximise S[f] while holding the first two moments fixed (mean μ, variance σ2), a straightforward Lagrange-multiplier calculation gives:
f⋆(x)=12π σ exp [−12(x−μσ)2],
the ordinary normal distribution.
In words: out of all distributions that share the same variance, the Gaussian has the largest entropy.
what it means is that the Gaussian is derived from Boltzmann’s entropy principle; the principle is not contained inside the Gaussian.
Your proposed alternative definition (“stationary distribution of binary-interacting elements of any closed system”) is much broader; it would in general not yield a Gaussian unless the interaction rules conserve only the first two moments.
3. What B-matrices (as described) would need to show
If the 2020 “B-matrix” formalism produces a Gaussian for any closed, binary-interaction system, then the method is implicitly assuming Boltzmann’s molecular-chaos postulate and restricting to interactions that leave only the mean and variance as conserved quantities. In that special case it reproduces the classical H-theorem pathway—from which Boltzmann’s entropy law originally came.
Bottom line
- Boltzmann entropy S=klnWS is a general variational principle.
- The Gaussian is the maximum-entropy solution when only μ and sigma^{2} are fixed.
- Therefore the Gaussian—and any “generalised Gaussian”—is derived from, not a superset for, Boltzmann’s law. Saying the law is a “subset” of a Gaussian puts the causal arrow backwards.
hope this makes sense.
best
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