In　this　paper,　we　apply　the　information　theory　to　provide　an　approximate　expression　of　the　steady-state　probability　distribution　for　blockchain　systems.　We　achieve　this　goal　by　maximizing　an　entropy　function　subject　to　specific　constraints.　These　constraints　are　based　on　some　prior　information,　including　the　average　numbers　of　transactions　in　the　block　and　the　transaction　pool,　respectively.　Furthermore,　we　use　some　numerical　experiments　to　analyze　how　the　key　factors　in　this　approximate　expression　depend　on　the　crucial　parameters　of　the　blockchain　system.　As　a　result,　this　approximate　expression　has　important　theoretical　significance　in　promoting　practical　applications　of　blockchain　technology.　At　the　same　time,　not　only　do　the　method　and　results　given　in　this　paper　provide　a　new　line　in　the　study　of　blockchain　queueing　systems,　but　they　also　provide　the　theoretical　basis　and　technical　support　for　how　to　apply　the　information　theory　to　the　investigation　of　blockchain　queueing　networks　and　stochastic　models　more　broadly.　©　The　Author(s),　under　exclusive　license　to　Springer　Nature　Switzerland　AG　2024.