You talk about macrostates. How do you define a macrostate for your initial configuration? Why this one and not another? Does it change over time?
Think of it from another angle. You have a box with some of these balls inside - all you know is that they all have the same energy. Do you agree that the macrostate won't change over time and the entropy will remain constant?
However, if you knew the positions they will be concentrated in some regions more than in others. That's what you called lower entropy in your example.
So what is it, does knowing more about a system change it's entropy or not?
Gibbs
>You talk about macrostates. How do you define a macrostate for your initial configuration? Why this one and not another? Does it change over time?
I'll choose something arbitrary. Temperature as measured by a thermometer is the macrostate. Every possible configuration of particles (microstates) that causes mercury to rise to a certain degree represents a different macrostate. And yes it changes with time; even at equilibrium.
>Think of it from another angle. You have a box with some of these balls inside - all you know is that they all have the same energy. Do you agree that the macrostate won't change over time and the entropy will remain constant?
No don't agree. If balls are located on one side of the box and the thermometer is on the other side of the box. The thermometer reads nothing. When the balls increase in entropy they collide with the thermometer producing a reading.
>However, if you knew the positions they will be concentrated in some regions more than in others. That's what you called lower entropy in your example.
Yeah I already don't agree with you, so the rest of your argument is gone. If you define macrostate as total energy in a system then yeah it never changes. But that's not the definition of macrostate. It's just one arbitrary choice you have chosen.
The more I think about your reply the less sense it makes to me.
What do you think is the definition of macrostate?
The standard notion in statistical mechanics is that if we have, for example, a volume of gas in equilibrium at (constant) ambient temperature and we measure the pressure it doesn’t change. The macrostate doesn’t change. That’s what being in equilibrium means. The macrostate is in that case defined by the variables P,T,V. If all you knew was the value of these three variables and they didn’t change how could the macrostate - or the entropy - change?
You tell me that if you know the precise position of the particles of that gas then the macrostate and the entropy change all the time.
But you also tell me that “Knowing more or less about a system does not change it's entropy.” Which is in flagrant contradiction with the two previous paragraphs.
I think the issue is more with your understanding then my explanation.
>What do you think is the definition of macrostate?
We first introduce the very fundamental statistical ideas of microstates and macrostates. Given a system (e.g., a gas), we view it as built from some elementary constituents, (e.g., molecules). Each constituent has a set of possible states it can be in. The thermodynamic state of the system (which characterizes the values of macroscopic observables such as energy, pressure, volume, etc. ) corresponds to many possible states of the constituents (the molecules). The collection of states of all the constituents is the microstate. To keep things clear, we refer to the macroscopic, thermodynamic state as the macrostate. The vast disparity between the number of possible macrostates versus microstates is at the heart of thermodynamic behavior! The number of distinct microstates giving the same macrostate is called the multiplicity of the macrostate. The multiplicity is a sort of micro-scopic observable which can be assigned to a macrostate.
>The standard notion in statistical mechanics is that if we have, for example, a volume of gas in equilibrium at (constant) ambient temperature and we measure the pressure it doesn’t change. The macrostate doesn’t change. That’s what being in equilibrium means. The macrostate is in that case defined by the variables P,T,V. If all you knew was the value of these three variables and they didn’t change how could the macrostate - or the entropy - change?
This notion is wrong. It CAN change. It just has an extreme low probability of changing from equilibrium to some low entropy state. The probability is low enough that you practically don't need to consider it, but you must consider it from a technical standpoint.
If by sheer luck all gas particles moved to the exact left side of the container. My measurement tool (thermometer) for macrostate was on the right side of the container then it registers zero. There is nothing in the laws of physics that prevents this from happening. Only probability makes this situation unlikely to happen.
>You tell me that if you know the precise position of the particles of that gas then the macrostate and the entropy change all the time.
No. I'm saying that the temperature reading on the thermometer a macroscopic measurement is INDEPENDENT of knowledge. Your mind doesn't control microstates and thus the macrostate of the system.
>But you also tell me that “Knowing more or less about a system does not change it's entropy.” Which is in flagrant contradiction with the two previous paragraphs.
Flagrant? You're offended? Well you can leave if you're offended. Using words like this also Offends me so we can end the conversation.
Ok. If you have a system at constant energy they are the same (all the microstates have the same probability). If you have a system a constant temperature a microstate can always correspond to different macrostates (it seems you’re not concerned about that ambiguity though).
In any case, the entropy is a property of (the ensemble of microstates that conform) the macrostate and not a property of the microstate. For a given macrostate you cannot use Gibbs entropy to talk about low-entropy and high-entropy microstates.
> Temperature as measured by a thermometer is the macrostate. Every possible configuration of particles (microstates) that causes mercury to rise to a certain degree represents a different macrostate.
How does your configuration where the balls where near one corner in the cube causes mercury to rise to a different level than the configuration where they occupy a larger volume near the center?
> And yes it changes with time; even at equilibrium.
What’s your definition of equilibrium? It’s hard for me to see what are we talking about, really.
> If balls are located on one side of the box and the thermometer is on the other side of the box. The thermometer reads nothing.
One could also say that if the balls are located anywhere within the box the thermometer reads nothing. The thermometer reading exists only when enough time has passed and the mercury-balls composite system is in equilibrium.
> When the balls increase in entropy they collide with the thermometer producing a reading.
Apparently you say that the entropy increases because the macrostate changes. But the macrostate is undefined until the balls hit the thermometer. But the entropy has to increase for the balls to move. It’s all a bit confusing, don’t you think?
Won’t the balls collide with the thermometer even if the entropy doesn’t change? It’s their movement that will take them there.
Consider your state - or any other state, for that matter - at t=0 and the state at t=1ps where the balls have advanced slightly. Is the macrostate different? How? Is the entropy different? How?
The balls have to touch thermometer. Otherwise the thermometer observes nothing. Deriving the exact formula of this interaction is complex but the intuition makes sense.
Interaction with the thermometer at a certain energy level produces a macrostate at Temperature T that is high probability. You see this macrostate throughout your lifetime. But nothing precludes the low probability macrostate (where all balls are at another corner of the box) from occuring. It's just that microstate has such a low probability of occuring you never see it. When nothing touches the thermometer, that temperature reading is the absolute zero macrostate.
>Apparently you say that the entropy increases because the macrostate changes. But the macrostate is undefined until the balls hit the thermometer. But the entropy has to increase for the balls to move. It’s all a bit confusing, don’t you think?
No. The macrostate is never undefined. I defined it as the measurement on that thermometer. Whatever you read on that thermometer (let's assume mercury levels move instantaneously and not be pedantic) then THAT is the current macrostate. I chose this set of macrostates, thus I pick that definition. There is no notion of undefined, the thermometer always HAS a reading.
>Won’t the balls collide with the thermometer even if the entropy doesn’t change? It’s their movement that will take them there.
They can't collide with the thermometer if they're on the other side of the box.
>Consider your state - or any other state, for that matter - at t=0 and the state at t=1ps where the balls have advanced slightly. Is the macrostate different? How? Is the entropy different? How?
Macrostate doesn't change if they're still far away from the thermometer. Still absolute zero. Only the microstate changed. The absolute zero macrostate includes all configurations of particles that will cause the thermometer to read 0. The probability of this macrostate occuring is quite low.
Entropy in this state is low. It increases as more and more particles interact with the thermometer.
The most probable series of events is that particles on the left corner of the box will begin to spread more evenly around the box. More and more particles will begin interacting with the thermometer causing the temperature to rise until it reaches some equilibrium. What can occur is all particles can by sheer luck suddenly concentrate to another corner of the box, but this is a low probability event.
To bring it back around to the main point. All of this is independent of your knowledge of the microstate. Your knowledge does not effect the outcome.
