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Mathematical Sciences Lab (Sakai)
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The lab rigorously proves results on phase transitions and critical phenomena in mathematics.
AboutAbout the lab
Rigorously proving how phase transitions work.
Prof. Sakai studies problems in mathematical physics, especially probability theory and statistical mechanics, with mathematical rigor.
The central interests are phase transitions and critical phenomena, and the limit theorems associated with them.
For example, the Ising model, a statistical-mechanical model of a magnet, shows a phase transition in which spontaneous magnetization appears or disappears as the temperature is changed.
Critical phenomena are thought to arise from the cooperation of infinitely many components, and they require a theory that goes beyond conventional probability theory dealing with independent variables.
Key terms
- Phase transition
- A sudden, drastic change in the properties of a substance when conditions such as temperature are changed.
- Critical phenomenon
- The anomalous behavior of various quantities near the very point where a phase transition occurs.
- Percolation
- A model of how water and the like seep through a random structure.
NewsLab news
View More ↗ResearchResearch themes
The lab studies problems in probability theory and statistical mechanics with mathematical rigor. Of particular interest are phase transitions and critical phenomena, and whether the limit theorems associated with them exist.
*Illustrative imagePhase transitions and critical phenomena
The Ising model, a statistical-mechanical model of a magnet, shows a phase transition in which spontaneous magnetization appears or disappears as the temperature is changed. Critical phenomena are the anomalous behavior of various quantities near the point of such a phase transition.
*Illustrative imageProbability theory beyond independence
Critical phenomena are thought to arise from the cooperation of the infinitely many components that make up a model. To truly understand them, a theory that goes beyond conventional probability theory dealing with independent variables is needed.
*Illustrative imageThe φ4 model and self-avoiding walks
The lab has studied the φ4 model, which appears in scalar field theory, and the self-avoiding walk, a statistical-mechanical model of polymers.
*Illustrative imagePercolation and the contact process
The lab has also worked on percolation, a model of the permeation process through a random structure; the contact process, a model of how an infectious disease spreads through society; and random walks that are influenced by their past history.
MembersFaculty
- ProfessorAkira Sakai
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “3Dpercolation” David Adams / CC BY 3.0 (cropped)
- Phase transitions and critical phenomena: “Homogeneous Nucleation Ising Model” Oerpli / CC BY-SA 3.0 (cropped)
- Probability theory beyond independence: “Textile cone (cropped)” Chiswick Chap / CC BY-SA 3.0 (cropped)
- The φ4 model and self-avoiding walks: “Polymer 2D” Alexis Monnerot-Dumaine / CC BY-SA 3.0 (cropped)
- Percolation and the contact process: “Front de percolation” Alexis Monnerot-Dumaine / CC BY-SA 3.0 (cropped)
- Related lab (Analysis (Kubo) Lab): “Surface waves” Roger McLassus / CC BY-SA 3.0 (cropped)
- Related lab (Furuhata Lab): “Canal-helix” Ag2gaeh / CC BY-SA 3.0 (cropped)
- Related lab (Human Mathematical Sciences Lab): “Diplora strigosa” LASZLO ILYES (laszlo-photo) from Cleveland, Ohio, USA / CC BY 2.0 (cropped)
- Related lab (Honda Lab): “Essential singularity” Functor Salad / CC BY-SA 3.0 (cropped)
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