Various concepts of limit in statistical physics

 

 

 

concept of limit

notations

• N : number of sites
• a : lattice spacing
• V : volume

continuum limit

• used in the lattice model
  
• sending the lattice spacing a to zero, and the number N of sites to infinity, while keeping the volume V=Na constant
  
• applied to spin chains whose continuum limit yields conformal field theories

scaling limit

• sounds similar to continuum limit
• sending the lattice spacing a to zero, while keeping the volume V and the correlation length fixed

thermodynamic limit

• increasing the volume together with the particle number so that the average particle number density remains constant.
  
• http://en.wikipedia.org/wiki/Thermodynamic_limit
  

infrared limit

• sending V to infinity, while keeping the lattice spacing a constant
  

ultraviolet limit

• ??
  

 

 

The c-theorem implies that the infra-red limit, where the scale goes to innity, and the ultra-violet limit, where the scale vanishes, are fixed points of the renormalisation group.

http://iopscience.iop.org/1126-6708/2000/03/008/pdf/1126-6708_2000_03_008.pdf

 

 

memo

• Glimm, J., Jaffe, A.: Particles and scaling for lattice fields and Ising models. Commun. Math. Phys.51, 1 (1976)
• Newman, C.M.: Normal fluctuations and the FKG inequalities. Commun. Math. Phys.74, 119 (1980)
• Fröhlich, J., Spencer, T.: Some recent rigorous results in the theory of phase transitions and critical phenomena. Séminaire Bourbaki No. 586 (February 1982)
• Sinai, Ya.G.: Mathematical foundations of the renormalization group method in statistical physics. In: Mathematical problems in theoretical physics. Dell'Antonio, G., Doplicher, S., Jona-Lasinio, G. (eds.). Lectures Notes in Physics, Vol. 80. Berlin, Heidelberg, New York: Springer 1978