Imagine mounting the camera of your phone to a microscope and recording the motion of dust particles dispersed in a glass of water. The motion of these particles (said to be “Brownian”) is nearly completely random, so that if we were to play the video forward or backwards we would not be able to tell the difference. Now imagine walking to a window overlooking a construction site and recording another video. This time we are clearly going to be able to distinguish forward from backward. This simple observation has very profound implications that are directly related to the so-called “arrow of time” and to how we can distinguishing living from nonliving systems.
In technical terms, we say that the motion of the dust particles obeys “time-reversal symmetry” (TRS) and that the evolution of a construction site breaks this symmetry. One of the defining properties of living systems is that they continuously consume energy to function and perform work. In this process they dissipate energy and inevitably break TRS. Likewise, in order for a building to be developed we must consume energy to do useful work (e.g., to assemble construction materials). If no energy was consumed and no work done, TRS would be preserved but nothing would get built.
A quantifiable signature of TRS breaking is a statistical quantity known as entropy production, which quantifies the cost of maintaining the system “alive”. Entropy production has been the subject of a great deal of interest in biophysics and it has traditionally been thought of as a single number that tells us whether the system obeys or breaks TRS. In this work, we introduce a methodology that decomposes entropy production into local spatial contributions - crucially, without knowledge of the underlying equations of motion. Put simply, imagine for instance tracking the motion of construction workers across different areas of the construction site, some will be more active than others, meaning that some areas will "produce" more entropy (i.e., break TRS more) than others. This new approach allows us to quantify which features of a system, and which events, are responsible for breaking TRS. Finally, we show that in the absence of local entropy production, we cannot extract work from the system from the features being tracked at that particular location in space.
We validated our approach in theory, simulation, and experiments of E. Coli swimming in the presence of obstacles. This approach should be applicable in a broad variety of domains. In biological systems we can probe the dynamics inside cells, their metabolic rates, and the microscopic motion from collective behavior of molecular motors. In micro through macroscopic fluid systems it will be interesting to study entropy production in vortices, near boundaries and at different scales through the turbulent regime. On a more macroscopic scale, traffic and crowds are amenable to our analysis as are communication and transportation systems and networks in general. And on yet larger scales one might consider entropy production at a cosmological level. We therefore expect that our technique will interest and provide a useful tool for researchers in a very wide variety of fields.