We propose a minimal coupled map lattice (CML) for the spiral pattern formation in astronomical objects which consist of accreting gas induced by gravity as a long-range force. In the proposed CML, we assume only two procedures: one in which the flow of gas particles occurs by gravity and another one in which the collision of gas particles occurs by advection in the flow. In spite of its simplicity, the numerical simulation of the proposed CML shows a new formation process in which grand design spiral patterns appear due to gas ejection from a central star. Several aspects of the formation process are indeed in good agreement with the results of conventional theories and observations. This agreement and the observations of outflows should suggest that the above gas ejection can happen and lead to the formation of grand design spiral patterns, in actual spiral galaxies and protoplanetary disks.

An entropy of the Ising model in the mean field approximation is derived by the Hamilton-Jacobi formalism. We consider a grand canonical ensemble with respect to the temperature and the external magnetic field. A cusp arises at the critical point, which shows a simple and new geometrical aspect of this model. In educational sense, this curve with a cusp helps students acquire a more intuitive view on statistical phase transitions.

The formation and disappearance of spiral arms are studied by focusing on jammed Keplerian gas in a coupled map lattice (CML) with a minimal set of procedures for simulating diverse patterns in astronomical objects. The CML shows that a spiral arm is a type of traffic jam, and its motion is governed by both a gas inflow into and outflow from the jam. In particular, we present a new approach to simply and directly evaluating the disappearance of spiral arms, called “light-in and heavy-out”. It is based on the gas flow rate difference between the light inflow and heavy outflow leading to the disappearance of traffic jams. Furthermore, we propose an approximate formula for the remaining lifetime of spiral arms, which is immediately derived from the “light-in and heavy-out” approach without calculating their pattern speeds as in conventional differential rotation. The proposed formula is successfully applied to the CML simulations.

The formation and disappearance of spiral arms are studied by focusing on jammed Keplerian gas in a coupled map lattice (CML) with a minimal set of procedures for simulating diverse patterns in astronomical objects. The CML shows that a spiral arm is a type of traffic jam, and its motion is governed by both a gas inflow into and outflow from the jam. In particular, a new mechanism for the disappearance of spiral arms is found. It is caused not by conventional differential rotation, but by the gas flow rate difference between the light inflow and heavy outflow, here called "light-in and heavy-out", leading to the disappearance of traffic jams. Furthermore, we propose a general approximate formula for the lifetime of spiral arms, which is simply derived from the mechanism of the "light-in and heavy-out". The proposed formula is successfully applied to the CML simulations, and moreover, to the observational data of the spiral galaxy M51.