We explore a link between complexity and physics for circuits of given functionality. Taking advantage of the connection between circuit counting problems and the derivation of ensembles in statistical mechanics, we tie the entropy of circuits of a given functionality and fixed number of gates to circuit complexity. We use thermodynamic relations to connect the quantity analogous to the equilibrium temperature to the exponent describing the exponential growth of the number of distinct functionalities as a function of complexity. This connection is intimately related to the finite compressibility of typical circuits. Finally, we use the thermodynamic approach to formulate a framework for the obfuscation of programs of arbitrary length -- an important problem in cryptography -- as thermalization through recursive mixing of neighboring sections of a circuit, which can viewed as the mixing of two containers with ``gases of gates''. This recursive process equilibrates the average complexity and leads to the saturation of the circuit entropy, while preserving functionality of the overall circuit. The thermodynamic arguments hinge on ergodicity in the space of circuits which we conjecture is limited to disconnected ergodic sectors due to fragmentation. The notion of fragmentation has important implications for the problem of circuit obfuscation as it implies that there are circuits with same size and functionality that cannot be connected via local moves. Furthermore, we argue that fragmentation is unavoidable unless the complexity classes NP and coNP coincide, a statement that implies the collapse of the polynomial hierarchy of complexity theory to its first level.

翻译：我们探索了给定功能电路在复杂度与物理之间的关联。利用电路计数问题与统计力学系综推导之间的联系，我们将给定功能及固定门数电路的熵与电路复杂度联系起来。通过热力学关系，将类似平衡温度的物理量与描述不同功能数量随复杂度指数增长的指数相关联。这种联系与典型电路的有限可压缩性密切相关。最后，我们采用热力学方法构建了一个适用于任意长度程序的混淆框架——这是密码学中的重要问题——该框架通过递归混合电路中相邻段实现热化，可视为两个装有"门气体"的容器混合过程。这种递归过程使平均复杂度达到平衡并导致电路熵饱和，同时保持整体电路的功能性。热力学论证依赖于电路空间中的遍历性，我们推测由于碎片化现象，该遍历性局限于不连通的遍历扇区。碎片化概念对电路混淆问题具有重要启示，因为它表明存在具有相同尺寸和功能却无法通过局部移动相连的电路。此外，我们论证除非复杂度类NP与coNP重合（这一论断意味着复杂度理论的多项式层级坍缩至第一层），否则碎片化现象不可避免。