That's a form of unkindness that I'll make it a point to push back against.

Sections of the principal bundle assign an element of a group to each point of a manifold. It's something like a group element field. Are there ever cases where we apply different group elements at different points in space? I thought we always just apply one group element globally to all points in spacetime (and to all fields).

Or do I misunderstand principal bundles (highly likely)

It's officially time for thermal physics ☕

yay

@Jagerber48 It's the gauge, the gauge can be anything

The typical example being the Lorenz gauge, which as you know is a solution to a differential equation

Lorenz gauge is some harmonic function

(also the principal bundle isn't assigning a group, it's a torsor, it's the difference between two gauges that is in a group)

@Jagerber48 sections of principal bundles are not very interesting because only the trivial principal bundle has global sections - non-trivial bundles do not have any

"Plato, who came next to them, caused mathematics in general and geometry in particular to make a very great advance, owing to his own 'zeal for these studies; for every one knows that he even filled his writings with mathematical discourses and strove on every occasion to arouse enthusiasm for mathematics in those who took up philosophy."

Truly he was the pop science man of his days

read about Plato's realm of being and realm of becoming. he was a visionary

im sure he didnt mean Platonism as literal universes. it is probably more nuanced

"Plato, of course, ignores him throughout his dialogues, and is said to have wished to burn all his works;"

Not a big fan of Democritus it seems

Democritus's work also seems correct

atomic theory is correct. maybe Plato didnt like the specifics of his reasoning

He most certainly did not like the generals of his reasoning either

it is just atomic theory. maybe plato didnt believe it then

Democritus has pioneered this theory after Leucippus

Some of Plato's opinions are nuanced, but others may be wrong

In this video around 19:30, the speaker (Suvrat Raju) says that Gauss law implies that the dS ground state is invariant under isometry group of dS space...How is Gauss law supposed to imply that?

He says that something like this is also done in electrodynamics---that Gauss law on the boundary says that the states are invariant?

does the 1/2 from $E=\frac{3}{2}k_B t_k$ come from $\frac{1}{2}mv^2$

Yes

that times the number of degrees of freedom

was listening to this and I wondered what he meant by "glitch", but yeah that makes sense.

I mean you could redefine masses to remove that term I guess

But then you'd have p = mv/2

he goes on to explain Carnot's "entropy" and how it's defined by energy/degrees kelvin but kelvin was invented afterwards. I wonder what Carnot worked in, celsius or fahrenheit

i think i understand the statistical definition of entropy but not sure how it's related to these definitions by boltzmann and sadi

Carnot was French :p

He's not gonna use fahrenheit

hmm boltzmann's entropy is $\frac{1}{k_B}S_{carnot}$ so it seems like that definition includes the probablilities of all the states in the phase space. Carnot's definition is just the conserved quantity of work done?

Actually, he only wrote "degrés".

@Obliv by "statistical definition" do you refer to the Shannon entropy?

technically Shannon entropy was defined after the thermodynamic entropy, but yeah it is the more fundamental concept imo

not sure, i'm just going off of this lecture series. for discrete phase space $-\sum_i P_i \log{P_i}$ and continuous $-\int P(p,q)\log{P(p,q)}\text{dp}\text{dq}$

@Obliv Yeah, that's Shannon entropy. Basically the more fundamental concept which is used in information theory

if boltzmann knew the value of his constant, he could relate temperature in "human" units to the energy of the system

I think boltzmann also used that definition

oh nvm, maybe not

It's been a while but iirc Boltzmann entropy is basically just a special case of shannon entropy

It also has a change of dimensions due to multiplying by the Boltzmann constant, which is more of a convention if anything

is thermal equilibrium when the probability density becomes uniform over the whole space for an open system?

@Obliv If you notice, you wrote $\mathrm{log}$ in that definition without specifying a base for the logarithm. That's because pretty much any base is valid for defining Shannon entropy, but it does change the units of your entropy

E.g. entropy using a base 2 logarithm would be in "bits", while entropy using a natural logarithm would be in "nats"

I think base 10 gives a "dit"

bits in base 2 would just get rid of the $\log{2}$ factor

so you'd have $S = n$

due to logarithm rules, a change in the base is the same as multiplying by a constant. so it is really a change of units at the end of the day

yeah

is anyone familiar with dynamical lie algebras