Thermodynamic properties and Hilbert space of human brain

Any macro system consists of a lot of micro particles. According to statistical physics the macro properties of a system are realized as the statistical average of the corresponding micro properties. In our study, a model based on ensemble theory from statistical physics is proposed. Specifically, the functional connectivity (FC) patterns confirmed by LEiDA are taken as the micro states of a system, and static functional connectivity (SFC) is seen as the macro state. When SFC can be written as the linear combination of these FC patterns, it is realized that these FC patterns are valid micro states of which the statistical results of relevant behaviors can describe the corresponding properties of SFC. In this case, the thermodynamic functions in ensemble theory are expressed in terms of theses micro states. We applied the model to study the biological effect of ketamine on brain, and proved by the {\it maximum work principles} that compared to the control group, the capability of work done of brain that has been injected ketamine has declined significantly. Moreover, the mechanic operator of brain is further studied and a Hilbert space which is spanned by the eigenvectors of the operator is gained. The confirmation of mechanic operator and Hilbert space has opened big probabilities with quantum mechanics of studying brain system, which would unfold a new era in relevant studies of neuroscience.