Critical Phenomena
Most of us are familiar with the three primary of phases matter: solid, liquid and gas. If you think about it, phase transitions are a mysterious and mystifying thing. Continuously raise the temperature of liquid water and you won't see anything change until a seemingly arbitrary threshold is surpassed (100 degrees Celsius) and then everything suddenly changes - the water boils! It seems there isn't a smooth transition between the liquid and gas phase of water because there aren't any intermediate steps between them. I can raise the temperature of ice by 50 degrees and nothing will happen to the ice, but raise the temperature one degree above 0 degrees Celsius and suddenly my ice is completely transformed into water with completely different properties!
Phase transitions are an example of critical phenomena – below a certain threshold one set of properties are exhibited and above a certain threshold a completely different set of properties are exhibited without any ground in between.
Let's reflect on how unusual this is. Boyle's law describes the relationship between the pressure and volume of a gas:
Here is a graph of the kinetic energy of a rotating sphere according to classical and relativistic physics:
Notice that both these graphs are continuous and smooth. The mathematical machinery of calculus was designed to describe and to study smooth changes – what mathematicians call continuous functions. Critical phenomena are profoundly discontinuous. Here is a basic graphical representation of phase transitions:
Notice that this graph is not smooth or continuous and you cannot take the derivative of this curve. This graph implies a completely different type of relationship between parameters than we're used to.
How do we make sense of this data? How do these coherent properties emerge from the simple interaction of particles on an atomic level? What mathematical techniques and models can account for these observations? Once we figure out how to model phase transition mathematically, we can abstract this analysis and think about other contexts in which the theory and mathematics of phase transition might be useful. After a lecture or performance there are a few moments of tension when some members of the crowd may stand up to offer the lecturer or performer a stranding ovation – at this point, one of two things can happen: 1) a handful of people will stand up then sit back down without precipitating a full stranding ovation, 2) everyone in the audience will follow the lead of the handful that stood up and offer a full standing ovation. There really isn't any middle ground in this scenario. Performers either get standing ovations or they don't, they don't get half a standing ovation. Like evaporating water, this can be thought of as a critical phenomena. A critical mass of awesomeness must be achieved by a performer to achieve a standing ovation in his audience.
So here are some questions to think about:
For the record: there are two other, more exotic phases of matter which happen at extreme temperatures, they are plasma and Bose-Einstein condensate. Plasma is by far the most abundant state of matter in our universe because it's the stuff stars are made out of and the latter is uniquely cool because it only happens at temperatures near absolute zero and it allows for quantum mechanical properties of matter to be manifest on large scales and be visible to the naked eye:
Phase transitions are an example of critical phenomena – below a certain threshold one set of properties are exhibited and above a certain threshold a completely different set of properties are exhibited without any ground in between.
Let's reflect on how unusual this is. Boyle's law describes the relationship between the pressure and volume of a gas:
Here is a graph of the kinetic energy of a rotating sphere according to classical and relativistic physics:
How do we make sense of this data? How do these coherent properties emerge from the simple interaction of particles on an atomic level? What mathematical techniques and models can account for these observations? Once we figure out how to model phase transition mathematically, we can abstract this analysis and think about other contexts in which the theory and mathematics of phase transition might be useful. After a lecture or performance there are a few moments of tension when some members of the crowd may stand up to offer the lecturer or performer a stranding ovation – at this point, one of two things can happen: 1) a handful of people will stand up then sit back down without precipitating a full stranding ovation, 2) everyone in the audience will follow the lead of the handful that stood up and offer a full standing ovation. There really isn't any middle ground in this scenario. Performers either get standing ovations or they don't, they don't get half a standing ovation. Like evaporating water, this can be thought of as a critical phenomena. A critical mass of awesomeness must be achieved by a performer to achieve a standing ovation in his audience.
So here are some questions to think about:
- How should we think about and describe threshold parameters?
- Why is the transition between the solid and liquid phases so sudden and discontinuous?
- What analogy can be drawn between phase transitions of matter and social phase transitions like standing ovations?
- What are some other examples of phase transition that aren't unique to matter?
For the record: there are two other, more exotic phases of matter which happen at extreme temperatures, they are plasma and Bose-Einstein condensate. Plasma is by far the most abundant state of matter in our universe because it's the stuff stars are made out of and the latter is uniquely cool because it only happens at temperatures near absolute zero and it allows for quantum mechanical properties of matter to be manifest on large scales and be visible to the naked eye:
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