Brownian Motion

J_B_ · Nov 17, 2021

essentialsaltes said:
I think a liquid is too dense to even consider B an outside possibility. A particle will have surface interactions with the liquid molecules that would prevent straightline motion.

If the 'particle' were itself an atom, and the fluid were a gas, then we might be in B territory.

Liquids have a wide range of densities & viscosities, so it may depend. It might also depend on the mass of the particle versus the forces applied by colliding particles, but fair enough. Would the motion of the particle in a dense, viscous liquid be more the "jiggling" that originally prompted curiosity about Brownian motion, or would it actually walk, just not in a straight line?

I like the possibility that one could derive conditions under which the particle wouldn't walk at all, but only jiggle in place. But it seems that if the particle can walk, then any path is possible. Or maybe I'm missing something in what you said.

HARK! · Nov 17, 2021

J_B_ said:
I like the possibility that one could derive conditions under which the particle wouldn't walk at all, but only jiggle in place.

Let's go back to the dice.

Would not 2,3,2,3,2,3 be jiggling in place?

FrumiousBandersnatch · Nov 17, 2021

HARK! said:
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.

No, he is right. The covalent water bonds are stronger than the ionic NaCl bonds, so the salt dissolves, meaning it separates into Na+ and Cl- ions. Electrolysis will give you chlorine & hydrogen gas, and sodium hydroxide: 2NaCl + 2H₂O → 2NaOH + H₂ + Cl₂

J_B_ · Nov 17, 2021

HARK! said:
Let's go back to the dice.

Would not 2,3,2,3,2,3 be jiggling in place?

Good point, if I'm understanding you. Jiggling in place is really no different than any other path. But that circles back to the straight line question. Is there a condition that would allow 2,3,2,3,2,3, but not 2,2,2,2,2,2?

HARK! · Nov 17, 2021

J_B_ said:
Good point, if I'm understanding you. Jiggling in place is really no different than any other path.

Exactly

J_B_ said:
Is there a condition that would allow 2,3,2,3,2,3, but not 2,2,2,2,2,2?

I can't think of one, unless we add other factors.

essentialsaltes · Nov 17, 2021

J_B_ said:
But it seems that if the particle can walk, then any path is possible.

If the particle can walk, it can travel as far as you like from where it starts (but not in a straight line). However, it might take a long time to get any considerable distance. Under typical idealizing assumptions, the average distance traveled goes as the square root of the number of 'steps'.

J_B_ · Nov 17, 2021

essentialsaltes said:
If the particle can walk, it can travel as far as you like from where it starts (but not in a straight line). However, it might take a long time to get any considerable distance. Under typical idealizing assumptions, the average distance traveled goes as the square root of the number of 'steps'.

Would it be correct, then, to summarize your position as saying there is a state that is not reachable? If so, can you repeat your reasons for that - formulate a proof, so to speak? I'm intrigued that there could be a state in a random system that is not reachable.

Hans Blaster · Nov 17, 2021

FrumiousBandersnatch said:
No, he is right. The covalent water bonds are stronger than the ionic NaCl bonds, so the salt dissolves, meaning it separates into Na+ and Cl- ions. Electrolysis will give you chlorine & hydrogen gas, and sodium hydroxide: 2NaCl + 2H₂O → 2NaOH + H₂ + Cl₂

As the illustration in the article you linked showed, the Na+ and Cl- ions gather clusters of water molecules, but don't fully bond to them.

Water itself also "autoionizes" (don't know if that's the word, I haven't taken chemistry in 30 years) with a fraction of water molecule pairs becoming OH- and H3O+ ions (again with neutral water clusters around them oriented by the charge). The pH of 7 for pure water represents the level of that autoionization. (1 part in 10^7 I think)

essentialsaltes · Nov 17, 2021

J_B_ said:
Would it be correct, then, to summarize your position as saying there is a state that is not reachable?

I don't think that's what I said.

I'm intrigued that there could be a state in a random system that is not reachable.

If it were a matter of an idealized random walk, then the answer is B.

However, an actual dust particle in a liquid will be subject to van der waals forces and other intermolecular forces that are real, but not part of the idealized model. So that the answer to the OP is option A. It's impossible.

HARK! · Nov 17, 2021

essentialsaltes said:
I don't think that's what I said.

If it were a matter of an idealized random walk, then the answer is B.

However, an actual dust particle in a liquid will be subject to van der waals forces and other intermolecular forces that are real, but not part of the idealized model. So that the answer to the OP is option A. It's impossible.

Would not these other forces be random too? If not, then in practicality, the effects of Brownian motion cannot yield a random result.

J_B_ · Nov 17, 2021

essentialsaltes said:
If it were a matter of an idealized random walk, then the answer is B.

OK.

essentialsaltes said:
However, an actual dust particle in a liquid will be subject to van der waals forces and other intermolecular forces that are real, but not part of the idealized model. So that the answer to the OP is option A. It's impossible.

OK. Has this been verified, or is it just your expectation?

Regardless, doesn't that mean certain states aren't reachable? I guess we'd have to list all the state variables:
* position
* momentum
* energy (or temperature?)
* ...

It seems that if the paths used to reach a position differ, then some energy state (or momentum?) would also differ. I'm not familiar with whether all the various intermolecular forces are conservative, but if a simple idea of friction is in play during the walk, then the length of the path would determine how much energy is dissipated. Given a straight line would be the shortest path, if it is impossible, that length (and the energy dissipated) couldn't be replicated by any other path.

Mark Quayle · Nov 17, 2021

HARK! said:
Would a NaCl molecule, in solution, be considered a particle, or a liquid?

I don't know... I expect it depends who you are talking to and what their context is.

Mark Quayle · Nov 17, 2021

J_B_ said:
The use of the term particle is always context sensitive.

I'll go you one step further. I'll bet if you saw a particle moving in a straight line through the fluid you would suspect something other than Brownian motion.

True that!

Edit: But then, let's say the particle was an amoeba, and was sloughing around or absorbing whatever obstacles were in its way —then maybe its center of mass following a perfectly straight line but not its edges! You would have non-Brownian center mass, with Brownian ...er no, because it would not be random.

essentialsaltes · Nov 17, 2021

HARK! said:
Would not these other forces be random too?

In a sense yes. But my objection is that they are continuous. The model that each 'kick' moves the particle in exactly one direction is not plausible, due to the influence of the environment.

essentialsaltes · Nov 17, 2021

J_B_ said:
Regardless, doesn't that mean certain states aren't reachable? I guess we'd have to list all the state variables:
* position
* momentum
* energy (or temperature?)

But you're not talking about a particular state, but an entire path.

HARK! · Nov 17, 2021

J_B_ said:
I'll go you one step further. I'll bet if you saw a particle moving in a straight line through the fluid you would suspect something other than Brownian motion.

If I was playing a dice game; and I rolled straight sixes, 100 times in a row; my opponents would suspect something other than random chance.

sjastro · Nov 17, 2021

Brownian motion can never be in a straight line and is modelled by the Langevin equation.

The first term in the right hand side of the equation is the resistance of a particle of mass m to the molecules in the medium and is proportional to the velocity of the particle v.
The proportionality constant λ is a damping coefficient.
The second term based on statistical mechanics is referred to as the noise term and represents the collisions of the air or liquid molecules with the particle.
The kinetic energy of the colliding molecules has a Gaussian distribution.

HARK! · Nov 17, 2021

essentialsaltes said:
But you're not talking about a particular state, but an entire path.

Plausible and possible are two different things. I would argue that one path, is just as possible as any other path, over the same distance.

J_B_ · Nov 17, 2021

essentialsaltes said:
But you're not talking about a particular state, but an entire path.

I may have muddled the question the first time, but I tried to clarify it to indicate a particular state.

Pick a starting point and an ending point. We know the state of the particle at the starting point. Is the state of the particle the same for every path when it reaches the ending point? I think not. In the simplest sense, the ending states would only be equal for paths of equal length ... the real case would be more complicated, but in general I think the idea would hold.

So, if a straight line is the shortest path, with the least dissipation of energy, no other path will duplicate the state at the end point. It is unreachable with respect to the given initial state.

J_B_ · Nov 17, 2021

HARK! said:
If I was playing a dice game; and I rolled straight sixes, 100 times in a row; my opponents would suspect something other than random chance.

Indeed. And yet the possibility still remains.

But dice may not be a sufficient analogy for what we're discussing, now that @essentialsaltes has taken us into the realm of real Brownian motion. And that's the discussion I'm more interested in.

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