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Entropy and the second law of thermodynamics have been credited with defining the arrow of time
predicting the ultimate heat death of the universe and providing the driving force for the development of structure as well as decay and
Also, excusing the messiness of your room, but what is entropy really and how fundamental is it to our universe?
Entropy is surely one of the most intriguing and misunderstood concepts in all of physics the entropy of the universe
Must always increase. So says the second law of thermodynamics. It's a law that seems emergent from deeper laws
It's statistical in nature and yet may ultimately be more fundamental and unavoidable than any other law in physics
Einstein said that the thermodynamics that encapsulates the second law is the only physical theory of universal content
Which I am convinced will never be overthrown and the great astrophysicists or Arthur Eddington
Warned if your theory is found to be against the second law of thermodynamics. I can give you no
Hope there is nothing for it but to collapse in deepest humiliation
We've looked at entropy in the past, but it's time to go much deeper to unravel the great unraveling
Over some upcoming episodes. We'll explore different aspects and consequences of entropy
Including its role in black hole thermodynamics and how it will lead to the end of our universe
but today we'll see what entropy really is and why the second law of thermodynamics is considered to be so fundamental and so
Unavoidable. Let's start from the beginning in 1824
Siddiqa, no published his reflections on the motive power of fire in which he revealed the theory for perfect engine efficiency
Heat engines which in Connors day were the newfangled steam engines work by turning the flow of heat energy into mechanical energy
for heat to flow you need two reservoirs of different temperature a
Perfectly efficient engine one undergoing the Carnot cycle converts all transferred heat energy into useful work in
Principle that work can then be converted back into heat? And so the temperature differential can be re-established
However, an inefficient engine will slowly deplete the difference in temperature reducing the heat flow and the engine whines down
Around a half century after Cano Rudolf Clausius was inspired to quantify this tendency of heat energies to decay over time
Enter entropy
Classiest define entropy as the internal property that changes as heat energy moves around within a system
Specifically the change in entropy of each reservoir is the heat energy going into or out of that reservoir
Divided by its temperature for a Carnot cycle the overall change in entropy is zero
but for any less efficient cycle entropy increases in
Fact an increase in entropy means that the heat reservoirs are approaching the same temperature
Reducing the capacity to do useful work
coño and Klaus uses were revealed entropy is a measure of how evenly spread out a systems energy is
the more evenly spread the less useful the energy is and
For an isolated system the best you can hope for is that the separation of energy and the entropy remain constant in?
reality
It will almost always increase unless energy comes in from the outside to reestablish the temperature differential
This understanding of entropy is in terms of flowing Heat and it came from the days when many including Cano himself
Belief that heat was a physical fluid called caloric. It took a revolution to understand the reality of entropy that
Revolution was statistical mechanics founded by the great Ludwig Boltzmann with his kinetic theory of gases
This theory explained thermodynamic behavior as the summed result of the individual motion of tiny particles
under Newton's laws of motion
Stat mech is really astounding it's founded on an absurdly simple idea
for a given set of large-scale observable properties every
Possible configuration of particles that could give those properties is equally likely
Let's add some physics speak by configuration
I mean the exact arrangement of positions velocities had said of all microscopic particles we call this the microstate and
We call the specific combination of large-scale macroscopic properties
The macro state macro states are entirely defined by thermodynamic properties
temperature pressure volume and number of particles
For a given macro state all microstates consistent with its thermodynamic properties are equally likely
For some macro States there are lots of different microstates or arrangements of particles that lead to roughly the same
Thermodynamic properties while other macro states can be produced by only very few
Microstates. Ok, one more fact if you leave a system to do its own thing
It'll eventually try out all possible microstates that are possible given the laws of physics
All particle arrangements will eventually happen
So if you look at the system at some random point in time
It'll be in a completely random microstate chosen from all possible microstates
And what macro state will have been well probably the one that's consistent with the most
Microstates we can think of these micro and macro States with an analogy. This is a go board
Let's say you place a hundred and eighty Blackstone's at random. Every possible specific arrangement is considered a microstate
Well, the overall shape of the distribution would be the macro state
there are nearly two times ten to the power of hundred and seven ways to arrange the pieces and
Almost all of them are pretty evenly mixed
So roughly all the same macro state some microstates are weird though
And they give different macro states because they're different average distributions. For example, there's one
Where all the stones are on one side?
That microstate is a factor of two times ten to the power of one hundred seven less likely than one of the many smoothly mixed
Microstates and the larger the board the less likely it is to end up in such a weird arrangement
for a roomful of ten to the power of
26 molecules of air the chance of getting all of the air molecules on one side of the room by chance
Is so small that it never happens. We've been talking a lot about particle position
But really that go board is an analogy for all possible combinations of all property position momentum spin
vibration really any degree of freedom that the system can have we call this space of properties a phase space and
Instead of particles being distributed through position space a microstate is really defined by how energy is distributed through phase space
the average distribution of individual particles in phase space
defines the thermodynamic properties of the system
That's why these similarly shaped distributions on the go board
Correspond to the same macro state while the clustered spread does not okay
so if you leave a system alone long enough, its
Particles and its energy will find its way into all the different forms that are possible the vast majority of possible
distributions of energy leave the system very close to a single macro state that's the state of thermal equilibrium in which
Energy is maximally spread out and temperature pressure density volume, etc. Have the values we expect from classical thermodynamics?
Statistical mechanics tells us why large-scale systems have the properties they do but what does this have to do with entropy?
Well, Ludwig Boltzmann figured that out too the Boltzmann equation
Tells us that entropy is the logarithm of the number of microstates
Consistent with the current macro state times the Boltzmann constant
So our smoothly spread our equilibrium go board has a high entropy and our clustered board has low entropy
by the way
There are certain special
microstates special arrangements of particles that look highly ordered but are still
Consistent with our high entropy macro state, for example, if we try to draw pictures or write words in phase space
This is where we get to a point of common confusion
Order is not the same thing as low entropy and the second law isn't always the tendency towards disorder in
thermodynamic entropy the only special arrangements of particles that change entropy are the ones that change the
Thermodynamic properties not the ones that spell out cuss words will mess up your room to get deeper into that
We'll need to talk about information entropy which will also need for black hole thermodynamics
And we'll take another episode
Okay
So the macro state that defines thermodynamic equilibrium is by definition the one with the most microstates
Which also means the maximum entropy?
Any system not in equilibrium must increase in entropy
Simply because at any future time its current microstate will most likely be one of the more common types of microstate
This is assuming you don't force the system from the outside
And it's possible to take each ghost stone and place it on a particular spot to construct a special microstate or to use a vacuum
Pump and a glass wall to move all of the air to one side of the room in both cases
You are reducing the number of accessible microstates
Which by definition must reduce entropy but to do so
You must introduce an external source of energy
Heat must flow between your system and the outside universe in a way that increases the entropy of the universe as a whole
statistical mechanics
Inevitably leads to entropy and the second law and it does so by something so fundamental and basic
That it's impossible to deny. It comes from counting the ways that energy can be distributed
The inevitability of the rise of entropy is as fundamental as counting
That's why Einstein and Eddington were so sure of it, but entropy is also
Statistical and emerges from the behavior of particles under the laws of motion
This is where the second law appears to add something new to the universe
Not seen in the more fundamental laws
It seems to add the arrow of time
See the laws of motion where the Newtonian or quantum mechanical don't care about the direction of time and yet the second law of thermodynamics
Clearly distinguishes between the past and the future
We talked a little about this in our episode on the physics of life
Where we saw how entropy drives both the increase and decay of complexity
It's almost like the concept of time is emergent and statistical just like entropy again
We'll delve deeper into this in the future, but for now, please
Be careful to keep your number of accessible microstates low
avoid thermal equilibrium and keep being that brilliant macro state that is you until I see you next week on
Space-time the field of statistical mechanics has given us some of the most profound
insights into the working of both the large scale and quantum realms
Stunningly its foundations are in probability theory to really get stat mech you have to get probability
Brilliant org has a great course on probability that solidly founded on exercise based learning
honestly, that's the only way to get math to do it check it out because we'll be doing more stat mech and delving deeper into
entropy in the future
Learning about physics is much more than facts and memorizing
When done right it gives you a whole new way to look at the universe itself
Brilliant math and science done right is proud to support space-time to learn more about brilliant. Go to brilliant org slash
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