User Tools

Site Tools


advanced_notions:symmetry_breaking:mermin-wagner_theorem

Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
advanced_notions:symmetry_breaking:mermin-wagner_theorem [2021/03/18 12:34]
orbital Added layman explanation
advanced_notions:symmetry_breaking:mermin-wagner_theorem [2021/03/18 12:36]
orbital
Line 11: Line 11:
 The Mermin-Wagner theorem explains, why the existence of phase transitions like Magnetization or Freezing depends on the dimension of the material undergoing the transition. Imagine water molecules close to the freezin point in the following configurations:​ The Mermin-Wagner theorem explains, why the existence of phase transitions like Magnetization or Freezing depends on the dimension of the material undergoing the transition. Imagine water molecules close to the freezin point in the following configurations:​
  
- ​- ​a one-dimensional chain +  * a one-dimensional chain 
- a two dimensional monolayer of watermolecules +  ​* ​a two dimensional monolayer of watermolecules 
- a cube +  ​* ​a cube 
-The molecules are moving around because of their thermal energy but also interact with each other due to hydrogen bonding. For water to freeze, the temperature has to be low enough such that the bonding between the molecules becomes strong enough to keep the molecules fixed at their place.+The molecules are moving around because of their thermal energy but also interact with each other due to hydrogen bonding. 
 +  
 +For water to freeze, the temperature has to be low enough such that the bonding between the molecules becomes strong enough to keep the molecules fixed at their place.
 This means that the number of water molecules that are neighbours determines how hard it is for a water molecule to move away from its position. This means that the number of water molecules that are neighbours determines how hard it is for a water molecule to move away from its position.
 We can imagine that there are more neighbours, the higher the dimension of the configuration is. We can imagine that there are more neighbours, the higher the dimension of the configuration is.
 Mermin-Wagner predicts that for certain problems there is a lower critical dimension ​ below which there are not enough neighbors to build up a strong enough bond such that all the water molecules "​freeze"​ at their position. This behaviour is also called cooperativity. Mermin-Wagner predicts that for certain problems there is a lower critical dimension ​ below which there are not enough neighbors to build up a strong enough bond such that all the water molecules "​freeze"​ at their position. This behaviour is also called cooperativity.
 +
 Water freezing is actually a very complex problem and the above explanation is only meant to simplify the visualization and should not be taken to be physically correct. Water freezing is actually a very complex problem and the above explanation is only meant to simplify the visualization and should not be taken to be physically correct.
 But for simple problems like the continuous Ising model it is possible to calculate this lower critical dimension to be .  ​ But for simple problems like the continuous Ising model it is possible to calculate this lower critical dimension to be .  ​
advanced_notions/symmetry_breaking/mermin-wagner_theorem.txt · Last modified: 2021/03/18 12:36 by orbital