Brownian Motion

[Petros2015]

It seems then, though it's unlikely, that one possible random path is a straight line to the edge.

Some form of the Golden Spiral wouldn't surprise me if it surfaced, but I'm not convinced that it would.

Nah.
Guess not.

 

[Petros2015]

It could be done with electrolysis

Why does this somehow sound familiar, in the Christian context?

 

[J_B_]

For the sake of simplicity, let's say the particle is not moving at the end. Then yes, because state variables only describe the state a particle (or system) is in at a moment. They do not carry their 'histories' with them.

Yes, I know it doesn't carry its history with it. For the state of the particle, I guess you're right.

I got tripped up because I was thinking of the energy dissipated during the walk - thinking more of the state of the whole system (particle + fluid), I suppose. Now I'm not sure of myself. Would the dissipated energy be a function of the path? I believe for simple friction it would be, but I'm not sure if that applies to the particle suspended in a fluid.

 

[HARK!]

Why does this somehow sound familiar, in the Christian context?

How does this relate to the OP?

 

[essentialsaltes]

Would the dissipated energy be a function of the path? I believe for simple friction it would be, but I'm not sure if that applies to the particle suspended in a fluid.

If the collisions are not perfectly elastic, it would have to be. And I think in a dense/viscous fluid that would have to be the case.

Again going to the opposite extreme of a perfect ideal gas, where your particle was one of the gas atoms. Sometimes it goes faster, sometimes slower, sometimes it's here, sometimes it's there, but the total temperature/thermal energy of the total system is constant (haha in the other idealistic assumption of the system being perfectly isolated from the rest of the universe). So in this case, the macro-state stays the same over time, no matter how far your particle goes. Any slip from those idealizations and losses would occur.

 

[J_B_]

Any slip from those idealizations and losses would occur.

OK. Sure. No energy created or destroyed. Maybe I'm slowly getting there.

Does that bring us to entropy, then? Do the different paths lead to different amounts of entropy? So ... would the work done over the shortest path (a straight line) be unique? And therefore produce the least entropy?

 

[Petros2015]

How does this relate to the OP?

Hmm. er...
It doesn't. My apologies, I will remove if you wish.

 

[essentialsaltes]

Does that bring us to entropy, then? Do the different paths lead to different amounts of entropy?

Certainly. I mean, think of our idealized heat engine. With every stroke of the piston, a certain amount of entropy is generated. But the system comes back to the same initial state at the end of each cycle. So the total entropy depends on how many cycles are performed. It's not determined by the initial and final states alone.

So ... would the work done over the shortest path (a straight line) be unique? And therefore produce the least entropy?

I would think so, again with the analogy of friction.

 

[sjastro]

Brownian movement is a diffusion process which increases entropy.
A collection of particles in a gaseous or liquid medium in its initial state is clustered.
The histogram or the distribution of the position of each particle xₓ is narrow as the standard deviation is small.

Brownian movement causes the cluster to spread out and the standard deviation increases with time.

Since entropy also increases there is a correlation between standard deviation and entropy.
Plotting the RMS of the standard deviation against time gives the following graph.

This graph is similar to the function σ(t) = k√t where k is a constant and is analogous to velocity while the derivative dσ/dt ≠ 0 is the acceleration.
Since F = m(dv/dt) is Newton’s second law this implies there is a “force” which pulls the particles in the cluster apart.
This is the entropic force which increases the entropy of the system.

 

[J_B_]

Certainly. I mean, think of our idealized heat engine. With every stroke of the piston, a certain amount of entropy is generated. But the system comes back to the same initial state at the end of each cycle. So the total entropy depends on how many cycles are performed. It's not determined by the initial and final states alone.



I would think so, again with the analogy of friction.

Yay! I dun a guud thunk in fisicks.

 

[J_B_]

... there is a correlation between standard deviation and entropy.

Interesting.

 

[sjastro]

Interesting.

Entropic forces come straight out of thermodynamics and the use of total differentials as defined in calculus.
The change in the internal energy dE of a system is defined by the equation;

dH and dV are the changes in enthalpy and volume respectively, P is the pressure which is assumed to be constant.
dH relates to the heat content of the system while PdV is the work done by or on the system.

The enthalpy change at some temperature T can be expressed in terms of entropy change dS according to the equation;

Hence

The total differentials dU and dS can be expressed as;

Therefore;

Equating the coefficient for dV gives the equation;

Since the pressure P is simply force per unit area this final equation tells us there are two forces at work, the first term on the right hand side of the equation is the entropic force, the second term is our usual definition of a force associated with collisions.

An ideal gas which is composed of point like particles engaging in elastic collisions results in a zero second term and is completely entropic in origin.
Entropic forces saved thermodynamics as physicists would have a hard time explaining how an ideal gas would exert pressure if the mechanism was purely based on collisions.
When T=0 which is absolute zero the entropic forces vanish and thermodynamics becomes a purely classical model.

Some physicists think gravity could be an entropic force.
Entropic gravity - Wikipedia

 

[Bob Crowley]

If we used something like the Lotto as an allegory, if we had to draw every single ball from 1 to 40 in that order by random processes, the chance that would happen would be the same as any other combination of the 40 balls.

That is 1/40! which is a very, very small number. Log (1/40!) = -47.911 unbounded. My maths is rusty these days, but I presume it would be a decimal number with 47 zeros after the decimal point before you got to the first digit.

If we consider there are billions if not trillions of liquid molecules in the container of fluid, the probability they will line up their momentum to push the particle in a straight line is infinitely remote.

 

[tas8831]

Nope. H2O will not break the molecular bond with the Na and Cl. It could be done with electrolysis through a semipermeable membrane ; but then that would consume water, leaving us with lye and bleach.

The American Chemical Society says differently, But what do they know...

https://www.acs.org/content/acs/en/...tives/water-everywhere/naci-dissociation.html