Neogenesis is a term among several which refers to the formation of life from non-life. In the pre-DNA scientific era, there was, at one time, a theory of "spontaneous generation", which stated that life could arise, under the correct conditions, from non-life. Louis Pasteur was one of several scientists who ultimately contributed to debunking the theory.
The current use of Neogenesis refers to the formation of life as we know it from the basic building blocks of organic molecules, such as amino acids by change and the input of energy from the environment. The current hurdle in the study of how life began on Earth, is that there is currently no workable theory to explain Neogenesis. The Miller Experiment was said to have furthered the study of Neogenesis. It was performed by a researcher named Stanley Miller. Miller’s brilliant experiment was said at the time to have created "life in a test-tube." Miller took a mixture of chemicals that would, he thought, have existed on a primitive Earth, put them in an enclosed spark chamber, and subjected them to electrical currents simulating lightning in a "primordial soup". He collected all sorts of organic molecules and even amino acids using only raw elemental materials.
The problem with Miller's procedure was that he controlled the conditions unrealistically. He isolated the newly-formed amino acids and other molecules from the harmful effects of the electrical discharge he was using to initiate their formation in the first place. The stringent requirements thus reiterated the difficulty with finding the origin of life.
Looking at Neogenesis in purely mathematical terms is also unhelpful. The statistical probability of the chance formation of organic molecules is low, but they do form. the combination of those molecules into more complex structures becomes even more unlikely. The formation of self-replicating molecules is essentially statistically impossible based on our current understanding of organic chemistry.
Even with the most optimum conditions, and given the widest latitude in assuming favorable conditions, there is yet another great hurdle - that of the cell. The cell is the simplest form of independent life known to mankind. Prions, virions, and viruses all depend on the mechanisms within the sells of the host organism to assist with their reproduction. The cell is almost unimaginably complex, containing a complicated selectively permeable cell membrane, lysosomes, Golgi bodies, mitochondria, ribosomes, RNA, DNA, a structured nucleus, and a host of other complex organelles. The replication of DNA depends on tens to hundreds of proteins from simple to complex. Each of these proteins must be present under the right conditions, simply for the process to occur.
Neither statistics nor mathematics nor experimentation has suggested a viable mechanism for Neogenesis. In fact, the more we know about life, the more difficult it becomes to imagine what could have bridged the gap between life and non-life. In the end, both those who believe in Neogenesis are left in the same position as those who believe in a Creator. Until some proof one way or another is found, both sides must accept their beliefs based purely on Faith.