Editing Path integral formulation

Jump to navigation Jump to search
Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.

The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.

Latest revision Your text
Line 19: Line 19:
where <math>P</math> is the Trotter number. In the Trotter limit, where <math>P \rightarrow \infty</math> these equations become exact. In the case where <math>P=1</math> these equations revert to a classical simulation. It has long been recognised that there is an isomorphism between this discretised quantum mechanical description, and the classical [[statistical mechanics]] of polyatomic fluids, in particular flexible ring molecules<ref>[http://dx.doi.org/10.1063/1.441588      David Chandler and Peter G. Wolynes "Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids", Journal of Chemical Physics '''74''' pp. 4078-4095 (1981)]</ref>, due to the periodic boundary conditions in imaginary time. It can be seen from the first term of the above equation that each particle <math>x_t</math> interacts with is neighbours <math>x_{t-1}</math> and <math>x_{t+1}</math> via a harmonic spring. The second term provides the internal potential energy.  
where <math>P</math> is the Trotter number. In the Trotter limit, where <math>P \rightarrow \infty</math> these equations become exact. In the case where <math>P=1</math> these equations revert to a classical simulation. It has long been recognised that there is an isomorphism between this discretised quantum mechanical description, and the classical [[statistical mechanics]] of polyatomic fluids, in particular flexible ring molecules<ref>[http://dx.doi.org/10.1063/1.441588      David Chandler and Peter G. Wolynes "Exploiting the isomorphism between quantum theory and classical statistical mechanics of polyatomic fluids", Journal of Chemical Physics '''74''' pp. 4078-4095 (1981)]</ref>, due to the periodic boundary conditions in imaginary time. It can be seen from the first term of the above equation that each particle <math>x_t</math> interacts with is neighbours <math>x_{t-1}</math> and <math>x_{t+1}</math> via a harmonic spring. The second term provides the internal potential energy.  


The following is a schematic for the interaction between atom <math>i</math> (green) and atom <math>j</math> (orange). Here we show the atoms having five  Trotter slices (<math>P=5</math>), forming what can be thought of as a "ring polymer molecule". The harmonic springs between Trotter slices are in yellow, and white/blue bonds represent the classical [[intermolecular pair potential]].
The following is a schematic for the interaction between atom <math>i</math> (green) and atom <math>j</math> (orange). Here we show the atoms having five  Trotter slices (<math>P=5</math>), forming what can be thought of as a "ring polymer molecule". The harmonic springs between Trotter slices are in yellow, and white/blue bonds represent the [[intermolecular pair potential]].


:{| border="1"
:{| border="1"
Please note that all contributions to SklogWiki are considered to be released under the Creative Commons Attribution Non-Commercial Share Alike (see SklogWiki:Copyrights for details). If you do not want your writing to be edited mercilessly and redistributed at will, then do not submit it here.
You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource. Do not submit copyrighted work without permission!

To edit this page, please answer the question that appears below (more info):

Cancel Editing help (opens in new window)