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A representation of the diffusion process in terms of Entropy and Macroscopic Information.
Cosmological expansion sets the rate at which the maximum possible entropy of the universe increases. The actual entropy of the Universe rises to reach up to the maximum entropy.
THE ARROW OF TIME
By Sowmya Kumar
Several theories have emerged to explain our perception of Time. In this series, we deal with the Thermodynamic arrow and the Historical arrow that are in apparent contradiction in their explanations of our perception of Time. The Thermodynamic arrow, based on the irrefutable second law of Thermodynamics, asserts that our sense of time is determined by the direction of increasing entropy or increasing disorder. The Historical arrow of time, on the contrary, explains our perception of time based on increasing information or increasing order. For example,evolutionarily, living organisms have become more and more organised and complex.
We are in search of a model for our Universe that resolves this conflict. To do that, we first gather a sense of what we mean by Entropy and Information, and seek a possible relation between these two quantities.
Entropy
The second law of Thermodynamics states that an isolated system always moves towards maximum entropy or the state of Thermodynamic equilibrium.
Now, we attempt to describe our diffusion example in terms of Entropy. In our example, while introducing the KMnO4 drop into the beaker, we create a departure from equilibrium by creating a concentration gradient of KMnO4 molecules in the beaker. In the final state of uniform concentration (or thermodynamic equilibrium), we thus expect maximum entropy to be attained by the system. Thus, in terms of entropy, the diffusion process proceeds from an initial state of low entropy towards higher entropy.
A representation of the diffusion process in terms of Entropy and Macroscopic Information.
Information
Let us revisit our diffusion example once more in the light of its initial conditions. To prescribe the initial conditions, we need information of the state of the system. Information relating to statistical properties such as Temperature, Pressure and Volume are called Macroscopic Information. The information relating to molecular attributes such as velocities and momentum is Microscopic Information.
We define our system as the drop of KMnO4 molecules and water molecules in the beaker. When the drop of KMnO4 is introduced in the left side of the beaker, we know the volume of the drop which is Macroscopic Information. As the molecules move and collide with other molecules, correlations emerge between their velocities. After diffusion has reached a steady state, we may be able to measure (within some uncertainty) the position and velocities (microscopic information) of KMnO4 molecules to describe the state of the system.
In the initial state, we had Macroscopic Information to describe our system and no Microscopic Information. In the final state, all information is available in terms of Microscopic Information only. Thus, during the process of diffusion, Macroscopic Information is converted to Microscopic Information.
The diffusion of KMnO4 molecules can be seen as proceeding towards decreasing Macroscopic Information or increasing Entropy. Entropy can thus be regarded as negative Macroscopic Information.
In terms of Macroscopic Information, the second law predicts that any isolated system that can initially be described only with Macroscopic Information proceeds towards a state of minimum Macroscopic Information.
Heat Death
The Heat Death is a proposition based on the belief that the Thermodynamic Arrow is the Master arrow that describes our sense of Time. It is the premise of the Heat Death argument that the initial state of the universe was a low entropy state or a state of maximum Macroscopic Information. The argumentation behind this proposition is that naturally occurring processes observed today tend to increase the entropy of the Universe. The Heat Death theory predicts that the Universe is ‘running down’ towards a state of thermodynamic equilibrium or an ultimate dead state where no processes can occur and anything and everything shall cease to exist.
However, it fails to explain satisfactorily, why we also see increasing complexity or order in the Universe as addressed by the Historical Arrow of Time.
Cosmological Arrow of Time
David Layzer, in his paper titled ‘the arrow of time’, argues that it is not necessary that the universe originated in a state of thermodynamic dis-equilibrium (low entropy state) and the universe need not be ‘running down’ to its eventual death. He proposes an alternate theory to explain the origin of time that also resolves the conflict between the thermodynamic and historical arrows of time.
The universe is undergoing an expansion. Cosmological expansion means that the distance between any two parts of the universe that are not bound by gravitational energy is increasing with time. This expansion of the Universe occurs at a finite rate. The maximum entropy of the Universe is determined by its current state and this is subject to change due to the cosmological expansion of the Universe. The rate at which the maximum entropy of the Universe changes is thus set by the rate of cosmological expansion.
On the other hand, the local temperature and density of the Universe also needs to change to correspond to the expanding state of the Universe. The rate at which the local temperatures and densities changes is governed by the rate of equilibration processes. The actual entropy of the Universe is determined by the local values of temperature and density. The rate of change in the actual entropy is thus determined by the rate of equilibration processes.
David Layzer proposes a hypothetical state in the beginning of the Universe such that the Universe is in local Thermodynamic equilibrium. The Universe expands from this hypothetical state. If the rate of equilibration processes is larger than the rate of cosmological expansion, then thermodynamic equilibrium is always maintained in the Universe and the Universe is always in the state of maximum entropy. If the rate of equilibration processes is lower than the rate of cosmic expansion, then there can be local departures from thermodynamic equilibrium that lead to formation of new information or order.
Cosmological expansion sets the rate at which the maximum possible entropy of the universe increases. The actual entropy of the Universe rises to reach up to the maximum entropy.
Such a speculation allows for the co-existence of the Thermodynamic arrow as well as the Historical arrow of Time. While the actual entropy of the Universe continues to increase (as governed by the second law of Thermodynamics), the rate of increase in the maximum possible entropy governed by the cosmic expansion may be higher, leading to the formation of astronomical systems and other complex beings. Thus, new information is continually generated in the universe.
An element of surprise
Laplace famously stated that if there was a demon that was well acquainted with all the laws of physics, had information of all the possible system interactions and had the ability to put all this data to analysis, then this demon could predict the future. What Laplace meant was that the future is completely deterministic and why it presents an element of surprise is due to our inability to collect all information and analyse it. To such a demon, all the world’s a stage. Men and women are mere players.
The work of David Layzer suggests that due to continuous generation of information, the universe itself never contains all the information of its future states. No demon or super computer can ever assimilate all the information since information is generated continually (like entropy). As conscious living organisms that directly perceive this Universe, we will always be surprised by the future.
Layzer, D. (1975). THE ARROW OF TIME. Scientific American, 233(6), 56-69. Retrieved January 9, 2021, from http://www.jstor.org/stable/24949962
Layzer, David. (1976). The Arrow of Time. The Astrophysical Journal. 206. 559-569. 10.1086/154413.