The topology, even with its abrupt decisions, focuses on the essence of a system
in the same way that a phase diagram does. A molecular system can be in the liquid
state or the solid state, and the transition between these two states is abrupt (in the
limit of an infinitely large system). A phase diagram shares the characteristics of a
topologically partitioned space: it has sharp boundaries and it ignores the geometrical details of the system. Indeed, a phase diagram looks beyond the exact
positions of the atoms in the molecules that make up a system; the atoms can
vibrate while the molecules can translate and rotate. In the same way, QCT looks
beyond the exact trajectories of the gradient paths but focuses on their connectivities, which are robust over large deformations of the gradient paths themselves.
A topological atom is then analogous to a phase. This is an example of how Nature
itself apparently imposes binary structures onto reality: it makes sense to say that a
piece of matter is either a liquid or a solid and the boundary between the two is
sharp.
There are more examples of sharp compartmentalisation in Nature. One of the
deepest examples is the architecture of thermodynamics, which discerns the system
and the surroundings. It is essential to the theoretical and practical functioning of
thermodynamics that a point in space either belongs to the system or to the surroundings. Any fuzzy partitioning or delay in decision would paralyse any thermodynamic calculations or predictions. Secondly, Life itself, this most complex of
structures, has organised and evolved under the very existence of sharp boundaries.
Due to its small size, a cell membrane is a relatively sharp boundary between the
cytoplasm and the extracellular space. Of course, the boundaries are open (under
the control of specialised proteins in the cell’s lipid membrane). The boundaries of a
topological atom are also open in that electrons can swirl through them.
A topological atom is a pattern, comparable to the shape of water as it rapidly
cascades over a rock in a river. From a distance, the water appears standing still in a
barely fluctuating shape but of course the water itself streams through the pattern.
Thirdly, at a higher level, human societies have also carved up the Earth’s space in
non-overlapping subspaces with sharp boundaries, called countries. When a territory is not allocated to a single clear “attractor” such as China, Pakistan or India, as
in the case of Kashmir, then a dispute arises, proving the inherent human nature of
partitioning land into non-overlapping sections. Further examples of binary statuses
are found in the legal atmosphere where one is either alive or dead, married or not,
or guilty or innocent. The question then remains why Chemistry is not the right
locale to propose non-overlapping partitioning. What is so intrinsically fuzzy about
atoms and electron densities that would prevent sharp boundaries? Is life or human
society perhaps less fuzzy?
At the very end of this section on the topological atom, and on the wider
topological approach with its fundamental characteristics and consequences, we put
the topology to rest and look at energy instead. Energy is a quantum mechanical
observable and the main question is how it can be partitioned. This is the topic of
the next section, where we forget about the gradient vector field of the electron
density, at least at the start.
34
P.L.A. Popelier
in the same way that a phase diagram does. A molecular system can be in the liquid
state or the solid state, and the transition between these two states is abrupt (in the
limit of an infinitely large system). A phase diagram shares the characteristics of a
topologically partitioned space: it has sharp boundaries and it ignores the geometrical details of the system. Indeed, a phase diagram looks beyond the exact
positions of the atoms in the molecules that make up a system; the atoms can
vibrate while the molecules can translate and rotate. In the same way, QCT looks
beyond the exact trajectories of the gradient paths but focuses on their connectivities, which are robust over large deformations of the gradient paths themselves.
A topological atom is then analogous to a phase. This is an example of how Nature
itself apparently imposes binary structures onto reality: it makes sense to say that a
piece of matter is either a liquid or a solid and the boundary between the two is
sharp.
There are more examples of sharp compartmentalisation in Nature. One of the
deepest examples is the architecture of thermodynamics, which discerns the system
and the surroundings. It is essential to the theoretical and practical functioning of
thermodynamics that a point in space either belongs to the system or to the surroundings. Any fuzzy partitioning or delay in decision would paralyse any thermodynamic calculations or predictions. Secondly, Life itself, this most complex of
structures, has organised and evolved under the very existence of sharp boundaries.
Due to its small size, a cell membrane is a relatively sharp boundary between the
cytoplasm and the extracellular space. Of course, the boundaries are open (under
the control of specialised proteins in the cell’s lipid membrane). The boundaries of a
topological atom are also open in that electrons can swirl through them.
A topological atom is a pattern, comparable to the shape of water as it rapidly
cascades over a rock in a river. From a distance, the water appears standing still in a
barely fluctuating shape but of course the water itself streams through the pattern.
Thirdly, at a higher level, human societies have also carved up the Earth’s space in
non-overlapping subspaces with sharp boundaries, called countries. When a territory is not allocated to a single clear “attractor” such as China, Pakistan or India, as
in the case of Kashmir, then a dispute arises, proving the inherent human nature of
partitioning land into non-overlapping sections. Further examples of binary statuses
are found in the legal atmosphere where one is either alive or dead, married or not,
or guilty or innocent. The question then remains why Chemistry is not the right
locale to propose non-overlapping partitioning. What is so intrinsically fuzzy about
atoms and electron densities that would prevent sharp boundaries? Is life or human
society perhaps less fuzzy?
At the very end of this section on the topological atom, and on the wider
topological approach with its fundamental characteristics and consequences, we put
the topology to rest and look at energy instead. Energy is a quantum mechanical
observable and the main question is how it can be partitioned. This is the topic of
the next section, where we forget about the gradient vector field of the electron
density, at least at the start.
34
P.L.A. Popelier