Living organisms are dissipative structures that consist of a regular pattern in matter and energy; the identity of the particular atoms and molecules from which they are ‘made up’ does not remain constant over time. However, they exist in a physical Universe that we also describe using the language of micro- scopic physics. How can the identity of a biological organism be described using terms from the microstate level? This question turns out to be a particular case of a more general problem: how should one construe the properties and identities of macroscopic objects in terms of physical microstates? Even more fundamentally, what mathematical model is appropriate for representing the state trajectories of objects that do not have a fixed identity?

Macroscopic objects appear to have well-defined phenomenal properties such as mass or shape at any particular point in time. During the period over which such an object maintains a unique identity, these properties can be modelled by variables in a dynamical system or stochastic process. The challenge is how to model the properties of macroscopic objects over time periods in which their identity is more ambiguous: when an object appears from nowhere, splits apart into multiple sub-objects, merges with one or more other objects, or disappears entirely. The following simple example will illustrate the problem. A pool of water is a macroscopic object that, at any particular point in time, appears to have well-defined phenomenal properties such as shape, mass, volume, and loca- tion. What is more, these properties appear to change over time in a predictable manner that we imagine could be (more or less) captured in a dynamical sys- tems model or stochastic process model at the macrostate level. However, the picture becomes more complicated when we consider that an individual pool of water may evaporate, split into two, or that two may merge into one: how is a trajectory of mass over time to be modelled in such cases?

I propose a general formal model that allows properties to be ascribed to dis- tinct individuals existing at a particular time, in a way that also accommodates inheritance relationships between individuals at different times. The traditional notion of a state trajectory as a sequence of variables forms a special case of this formulation, in which there is a unique line of inheritance.